Theorem moexexvw 2620

Description: "At most one" double quantification. Version of moexexv 2631 with an additional disjoint variable condition, which does not require ax-13 2367. (Contributed by NM, 26-Jan-1997.) (Revised by Gino Giotto, 22-Aug-2023.) Factor out common proof lines with moexex 2630. (Revised by Wolf Lammen, 2-Oct-2023.)

Assertion

Ref	Expression
moexexvw	⊢ ((∃*𝑥𝜑 ∧ ∀𝑥∃*𝑦𝜓) → ∃*𝑦∃𝑥(𝜑 ∧ 𝜓))

Distinct variable groups:   𝑥,𝑦   𝜑,𝑦

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

Proof of Theorem moexexvw

Step	Hyp	Ref	Expression
1		nfv 1910	. 2 ⊢ Ⅎ𝑦𝜑
2		nfv 1910	. 2 ⊢ Ⅎ𝑦∃*𝑥𝜑
3		nfe1 2140	. . 3 ⊢ Ⅎ𝑥∃𝑥(𝜑 ∧ 𝜓)
4	3	nfmov 2550	. 2 ⊢ Ⅎ𝑥∃*𝑦∃𝑥(𝜑 ∧ 𝜓)
5	1, 2, 4	moexexlem 2618	1 ⊢ ((∃*𝑥𝜑 ∧ ∀𝑥∃*𝑦𝜓) → ∃*𝑦∃𝑥(𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1532  ∃wex 1774  ∃*wmo 2528

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-10 2130  ax-11 2147  ax-12 2167

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-nf 1779  df-mo 2530

This theorem is referenced by:  mosub  3707  funco  6588  tfsconcatlem  42756

Otomatik - 173.255.232.114 CloudFlare DNS Türk Telekom DNS Google DNS Open DNS

OSZAR »