Theorem moexex 2723

Description: "At most one" double quantification. Usage of this theorem is discouraged because it depends on ax-13 2390. Use the version moexexvw 2713 when possible. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 28-Dec-2018.) Factor out common proof lines with moexexvw 2713. (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 2646	. 2 ⊢ Ⅎ𝑦∃*𝑥𝜑
3		nfe1 2154	. . 3 ⊢ Ⅎ𝑥∃𝑥(𝜑 ∧ 𝜓)
4	3	nfmo 2646	. 2 ⊢ Ⅎ𝑥∃*𝑦∃𝑥(𝜑 ∧ 𝜓)
5	1, 2, 4	moexexlem 2711	1 ⊢ ((∃*𝑥𝜑 ∧ ∀𝑥∃*𝑦𝜓) → ∃*𝑦∃𝑥(𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 398  ∀wal 1535  ∃wex 1780  Ⅎwnf 1784  ∃*wmo 2620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-11 2161  ax-12 2177  ax-13 2390

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622

This theorem is referenced by:  moexexv  2724  2moswap  2729