Theorem 2moswap 2534

Description: A condition allowing swap of "at most one" and existential quantifiers. (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 2013	. . . 4 ⊢ Ⅎ𝑦∃𝑦𝜑
2	1	moexex 2528	. . 3 ⊢ ((∃*𝑥∃𝑦𝜑 ∧ ∀𝑥∃*𝑦𝜑) → ∃*𝑦∃𝑥(∃𝑦𝜑 ∧ 𝜑))
3	2	expcom 449	. 2 ⊢ (∀𝑥∃*𝑦𝜑 → (∃*𝑥∃𝑦𝜑 → ∃*𝑦∃𝑥(∃𝑦𝜑 ∧ 𝜑)))
4		19.8a 2038	. . . . 5 ⊢ (𝜑 → ∃𝑦𝜑)
5	4	pm4.71ri 662	. . . 4 ⊢ (𝜑 ↔ (∃𝑦𝜑 ∧ 𝜑))
6	5	exbii 1763	. . 3 ⊢ (∃𝑥𝜑 ↔ ∃𝑥(∃𝑦𝜑 ∧ 𝜑))
7	6	mobii 2480	. 2 ⊢ (∃*𝑦∃𝑥𝜑 ↔ ∃*𝑦∃𝑥(∃𝑦𝜑 ∧ 𝜑))
8	3, 7	syl6ibr 240	1 ⊢ (∀𝑥∃*𝑦𝜑 → (∃*𝑥∃𝑦𝜑 → ∃*𝑦∃𝑥𝜑))

Syntax hints:   → wi 4   ∧ wa 382  ∀wal 1472  ∃wex 1694  ∃*wmo 2458

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233

This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-eu 2461  df-mo 2462

This theorem is referenced by:  2euswap  2535  2rmoswap  39630