Theorem moexexvw 2629

Description: "At most one" double quantification. Version of moexexv 2640 with an additional disjoint variable condition, which does not require ax-13 2371. (Contributed by NM, 26-Jan-1997.) (Revised by Gino Giotto, 22-Aug-2023.) Factor out common proof lines with moexex 2639. (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 1922	. 2 ⊢ Ⅎ𝑦𝜑
2		nfv 1922	. 2 ⊢ Ⅎ𝑦∃*𝑥𝜑
3		nfe1 2153	. . 3 ⊢ Ⅎ𝑥∃𝑥(𝜑 ∧ 𝜓)
4	3	nfmov 2559	. 2 ⊢ Ⅎ𝑥∃*𝑦∃𝑥(𝜑 ∧ 𝜓)
5	1, 2, 4	moexexlem 2627	1 ⊢ ((∃*𝑥𝜑 ∧ ∀𝑥∃*𝑦𝜓) → ∃*𝑦∃𝑥(𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1541  ∃wex 1787  ∃*wmo 2537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-10 2143  ax-11 2160  ax-12 2177

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-nf 1792  df-mo 2539

This theorem is referenced by:  mosub  3615  funco  6398