Theorem moexex 2666

Description: "At most one" double quantification. Usage of this theorem is discouraged because it depends on ax-13 2404. Use the version moexexvw 2656 when possible. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 28-Dec-2018.) Factor out common proof lines with moexexvw 2656. (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 2590	. 2 ⊢ Ⅎ𝑦∃*𝑥𝜑
3		nfe1 2185	. . 3 ⊢ Ⅎ𝑥∃𝑥(𝜑 ∧ 𝜓)
4	3	nfmo 2590	. 2 ⊢ Ⅎ𝑥∃*𝑦∃𝑥(𝜑 ∧ 𝜓)
5	1, 2, 4	moexexlem 2654	1 ⊢ ((∃*𝑥𝜑 ∧ ∀𝑥∃*𝑦𝜓) → ∃*𝑦∃𝑥(𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 400  ∀wal 1568  ∃wex 1809  Ⅎwnf 1813  ∃*wmo 2565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-10 2176  ax-11 2192  ax-12 2213  ax-13 2404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-nf 1814  df-mo 2567

This theorem is referenced by:  moexexv  2667  2moswap  2672