Theorem 2moswap 2671

Description: A condition allowing to swap an existential quantifier and at at-most-one quantifier. Usage of this theorem is discouraged because it depends on ax-13 2403. Use the weaker 2moswapv 2656 when possible. (Contributed by NM, 10-Apr-2004.) (New usage is discouraged.)

Assertion

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

Proof of Theorem 2moswap

Step	Hyp	Ref	Expression
1		nfe1 2184	. . . 4 ⊢ Ⅎ𝑦∃𝑦𝜑
2	1	moexex 2665	. . 3 ⊢ ((∃*𝑥∃𝑦𝜑 ∧ ∀𝑥∃*𝑦𝜑) → ∃*𝑦∃𝑥(∃𝑦𝜑 ∧ 𝜑))
3	2	expcom 418	. 2 ⊢ (∀𝑥∃*𝑦𝜑 → (∃*𝑥∃𝑦𝜑 → ∃*𝑦∃𝑥(∃𝑦𝜑 ∧ 𝜑)))
4		19.8a 2216	. . . . 5 ⊢ (𝜑 → ∃𝑦𝜑)
5	4	pm4.71ri 569	. . . 4 ⊢ (𝜑 ↔ (∃𝑦𝜑 ∧ 𝜑))
6	5	exbii 1877	. . 3 ⊢ (∃𝑥𝜑 ↔ ∃𝑥(∃𝑦𝜑 ∧ 𝜑))
7	6	mobii 2575	. 2 ⊢ (∃*𝑦∃𝑥𝜑 ↔ ∃*𝑦∃𝑥(∃𝑦𝜑 ∧ 𝜑))
8	3, 7	imbitrrdi 255	1 ⊢ (∀𝑥∃*𝑦𝜑 → (∃*𝑥∃𝑦𝜑 → ∃*𝑦∃𝑥𝜑))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400  ∀wal 1567  ∃wex 1808  ∃*wmo 2564

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-10 2175  ax-11 2191  ax-12 2212  ax-13 2403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1572  df-ex 1809  df-nf 1813  df-mo 2566

This theorem is used by:  2euswap  2672