Theorem 2moswap 2547

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 2027	. . . 4 ⊢ Ⅎ𝑦∃𝑦𝜑
2	1	moexex 2541	. . 3 ⊢ ((∃*𝑥∃𝑦𝜑 ∧ ∀𝑥∃*𝑦𝜑) → ∃*𝑦∃𝑥(∃𝑦𝜑 ∧ 𝜑))
3	2	expcom 451	. 2 ⊢ (∀𝑥∃*𝑦𝜑 → (∃*𝑥∃𝑦𝜑 → ∃*𝑦∃𝑥(∃𝑦𝜑 ∧ 𝜑)))
4		19.8a 2052	. . . . 5 ⊢ (𝜑 → ∃𝑦𝜑)
5	4	pm4.71ri 665	. . . 4 ⊢ (𝜑 ↔ (∃𝑦𝜑 ∧ 𝜑))
6	5	exbii 1774	. . 3 ⊢ (∃𝑥𝜑 ↔ ∃𝑥(∃𝑦𝜑 ∧ 𝜑))
7	6	mobii 2493	. 2 ⊢ (∃*𝑦∃𝑥𝜑 ↔ ∃*𝑦∃𝑥(∃𝑦𝜑 ∧ 𝜑))
8	3, 7	syl6ibr 242	1 ⊢ (∀𝑥∃*𝑦𝜑 → (∃*𝑥∃𝑦𝜑 → ∃*𝑦∃𝑥𝜑))

Syntax hints:   → wi 4   ∧ wa 384  ∀wal 1481  ∃wex 1704  ∃*wmo 2471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-eu 2474  df-mo 2475

This theorem is referenced by:  2euswap  2548  2rmoswap  41184