Theorem 2euswap 1065

Description: A condition allowing swap of uniqueness and existential quantifiers.

Assertion

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

Proof of Theorem 2euswap

Step	Hyp	Ref	Expression
1		excomim 727	. . . 4 ⊢ (∃x∃yφ → ∃y∃xφ)
2	1	a1i 7	. . 3 ⊢ (∀x∃*yφ → (∃x∃yφ → ∃y∃xφ))
3		2moswap 1064	. . 3 ⊢ (∀x∃*yφ → (∃*x∃yφ → ∃*y∃xφ))
4	2, 3	anim12d 431	. 2 ⊢ (∀x∃*yφ → ((∃x∃yφ ∧ ∃*x∃yφ) → (∃y∃xφ ∧ ∃*y∃xφ)))
5		eu5 1035	. 2 ⊢ (∃!x∃yφ ↔ (∃x∃yφ ∧ ∃*x∃yφ))
6		eu5 1035	. 2 ⊢ (∃!y∃xφ ↔ (∃y∃xφ ∧ ∃*y∃xφ))
7	4, 5, 6	3imtr4g 426	1 ⊢ (∀x∃*yφ → (∃!x∃yφ → ∃!y∃xφ))

Syntax hints:   → wi 2   ∧ wa 196  ∀wal 672  ∃wex 678  ∃!weu 1007  ∃*wmo 1008

This theorem is referenced by:  2reuswap 1341  euxfr2 1435

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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