Theorem 19.3v 1978

Description: Version of 19.3 2191 with a disjoint variable condition, requiring fewer axioms. Any formula can be universally quantified using a variable which it does not contain. See also 19.9v 1980. (Contributed by Anthony Hart, 13-Sep-2011.) Remove dependency on ax-7 2004. (Revised by Wolf Lammen, 4-Dec-2017.) (Proof shortened by Wolf Lammen, 20-Oct-2023.)

Assertion

Ref	Expression
19.3v	⊢ (∀𝑥𝜑 ↔ 𝜑)

Distinct variable group:   𝜑,𝑥

Proof of Theorem 19.3v

Step	Hyp	Ref	Expression
1		spvw 1977	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
2		ax-5 1906	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
3	1, 2	impbii 208	1 ⊢ (∀𝑥𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 205  ∀wal 1532

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

This theorem depends on definitions:  df-bi 206  df-ex 1775

This theorem is referenced by:  19.27v  1986  19.28v  1987  19.37v  1988  axrep1  5291  axrep6  5297  kmlem14  10206  zfcndrep  10657  zfcndpow  10659  zfcndac  10662  lfuhgr3  34947  bj-snsetex  36670  iooelexlt  37069  dford4  42687  relexp0eq  43368

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