Theorem moexexvw 2655

Description: "At most one" double quantification. Version of moexexv 2666 with an additional disjoint variable condition, which does not require ax-13 2403. (Contributed by NM, 26-Jan-1997.) (Revised by GG, 22-Aug-2023.) Factor out common proof lines with moexex 2665. (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 1947	. 2 ⊢ Ⅎ𝑦𝜑
2		nfv 1947	. 2 ⊢ Ⅎ𝑦∃*𝑥𝜑
3		nfe1 2187	. . 3 ⊢ Ⅎ𝑥∃𝑥(𝜑 ∧ 𝜓)
4	3	nfmov 2587	. 2 ⊢ Ⅎ𝑥∃*𝑦∃𝑥(𝜑 ∧ 𝜓)
5	1, 2, 4	moexexlem 2653	1 ⊢ ((∃*𝑥𝜑 ∧ ∀𝑥∃*𝑦𝜓) → ∃*𝑦∃𝑥(𝜑 ∧ 𝜓))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1568  ∃wex 1812  ∃*wmo 2564

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2178  ax-11 2194  ax-12 2215

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-mo 2566

This theorem is used by:  mosub  3674  funco  6577  tfsconcatlem  44162