Theorem 2moswap 2674

Description: A condition allowing to swap an existential quantifier and at at-most-one quantifier. (Contributed by NM, 10-Apr-2004.)

Assertion

Ref	Expression
2moswap	⊢ (∀𝑥∃*𝑦𝜑 → (∃*𝑥∃𝑦𝜑 → ∃*𝑦∃𝑥𝜑))

Proof of Theorem 2moswap

Step	Hyp	Ref	Expression
1		nfe1 2144	. . . 4 ⊢ Ⅎ𝑦∃𝑦𝜑
2	1	moexex 2668	. . 3 ⊢ ((∃*𝑥∃𝑦𝜑 ∧ ∀𝑥∃*𝑦𝜑) → ∃*𝑦∃𝑥(∃𝑦𝜑 ∧ 𝜑))
3	2	expcom 404	. 2 ⊢ (∀𝑥∃*𝑦𝜑 → (∃*𝑥∃𝑦𝜑 → ∃*𝑦∃𝑥(∃𝑦𝜑 ∧ 𝜑)))
4		19.8a 2166	. . . . 5 ⊢ (𝜑 → ∃𝑦𝜑)
5	4	pm4.71ri 556	. . . 4 ⊢ (𝜑 ↔ (∃𝑦𝜑 ∧ 𝜑))
6	5	exbii 1892	. . 3 ⊢ (∃𝑥𝜑 ↔ ∃𝑥(∃𝑦𝜑 ∧ 𝜑))
7	6	mobii 2563	. 2 ⊢ (∃*𝑦∃𝑥𝜑 ↔ ∃*𝑦∃𝑥(∃𝑦𝜑 ∧ 𝜑))
8	3, 7	syl6ibr 244	1 ⊢ (∀𝑥∃*𝑦𝜑 → (∃*𝑥∃𝑦𝜑 → ∃*𝑦∃𝑥𝜑))

Syntax hints:   → wi 4   ∧ wa 386  ∀wal 1599  ∃wex 1823  ∃*wmo 2549

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-mo 2551

This theorem is referenced by:  2euswap  2675  2rmoswap  42149