Theorem 2moswap 1064

Description: A condition allowing swap of "at most one" and existential quantifiers.

Assertion

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

Proof of Theorem 2moswap

Step	Hyp	Ref	Expression
1		hbe1 709	. . . . 5 ⊢ (∃yφ → ∀y∃yφ)
2	1	moexex 1058	. . . 4 ⊢ ((∃*x∃yφ ∧ ∀x∃*yφ) → ∃*y∃x(∃yφ ∧ φ))
3	2	exp 291	. . 3 ⊢ (∃*x∃yφ → (∀x∃*yφ → ∃*y∃x(∃yφ ∧ φ)))
4	3	com12 13	. 2 ⊢ (∀x∃*yφ → (∃*x∃yφ → ∃*y∃x(∃yφ ∧ φ)))
5		19.8a 712	. . . . 5 ⊢ (φ → ∃yφ)
6	5	pm4.71ri 484	. . . 4 ⊢ (φ ↔ (∃yφ ∧ φ))
7	6	biex 733	. . 3 ⊢ (∃xφ ↔ ∃x(∃yφ ∧ φ))
8	7	bimo 1031	. 2 ⊢ (∃*y∃xφ ↔ ∃*y∃x(∃yφ ∧ φ))
9	4, 8	syl6ibr 186	1 ⊢ (∀x∃*yφ → (∃*x∃yφ → ∃*y∃xφ))

Syntax hints:   → wi 2   ∧ wa 196  ∀wal 672  ∃wex 678  ∃*wmo 1008

This theorem is referenced by:  2euswap 1065  2eu1 1067

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010

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