Theorem moexex 2668

Description: "At most one" double quantification. Usage of this theorem is discouraged because it depends on ax-13 2406. Use the version moexexvw 2658 when possible. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 28-Dec-2018.) Factor out common proof lines with moexexvw 2658. (Revised by Wolf Lammen, 2-Oct-2023.) (New usage is discouraged.)

Hypothesis

Ref	Expression
moexex.1	⊢ Ⅎ𝑦𝜑

Assertion

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

Proof of Theorem moexex

Step	Hyp	Ref	Expression
1		moexex.1	. 2 ⊢ Ⅎ𝑦𝜑
2	1	nfmo 2592	. 2 ⊢ Ⅎ𝑦∃*𝑥𝜑
3		nfe1 2188	. . 3 ⊢ Ⅎ𝑥∃𝑥(𝜑 ∧ 𝜓)
4	3	nfmo 2592	. 2 ⊢ Ⅎ𝑥∃*𝑦∃𝑥(𝜑 ∧ 𝜓)
5	1, 2, 4	moexexlem 2656	1 ⊢ ((∃*𝑥𝜑 ∧ ∀𝑥∃*𝑦𝜓) → ∃*𝑦∃𝑥(𝜑 ∧ 𝜓))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1568  ∃wex 1812  Ⅎwnf 1816  ∃*wmo 2567

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 2179  ax-11 2195  ax-12 2216  ax-13 2406

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 2569

This theorem is used by:  moexexv  2669  2moswap  2674