Theorem cbvald 2402

Description: Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 2446. Usage of this theorem is discouraged because it depends on ax-13 2367. See cbvaldw 2330 for a version with 𝑥, 𝑦 disjoint, not depending on ax-13 2367. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) (Revised by Wolf Lammen, 13-May-2018.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cbvald.1	⊢ Ⅎ𝑦𝜑
cbvald.2	⊢ (𝜑 → Ⅎ𝑦𝜓)
cbvald.3	⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))

Assertion

Ref	Expression
cbvald	⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Distinct variable groups:   𝜑,𝑥   𝜒,𝑥

Allowed substitution hints:   𝜑(𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑦)

Proof of Theorem cbvald

Step	Hyp	Ref	Expression
1		nfv 1910	. 2 ⊢ Ⅎ𝑥𝜑
2		cbvald.1	. 2 ⊢ Ⅎ𝑦𝜑
3		cbvald.2	. 2 ⊢ (𝜑 → Ⅎ𝑦𝜓)
4		nfvd 1911	. 2 ⊢ (𝜑 → Ⅎ𝑥𝜒)
5		cbvald.3	. 2 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))
6	1, 2, 3, 4, 5	cbv2 2398	1 ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1532  Ⅎwnf 1778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-11 2147  ax-12 2167  ax-13 2367

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-ex 1775  df-nf 1779

This theorem is referenced by:  cbvexd  2403  cbvaldva  2404  axextnd  10614  axrepndlem1  10615  axunndlem1  10618  axpowndlem2  10621  axpowndlem3  10622  axpowndlem4  10623  axregndlem2  10626  axregnd  10627  axinfnd  10629  axacndlem5  10634  axacnd  10635  axextdist  35395  distel  35399  wl-sb8eut  37045

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