Theorem excomim 727

Description: One direction of Theorem 19.11 of [Margaris] p. 89.

Assertion

Ref	Expression
excomim	⊢ (∃x∃yφ → ∃y∃xφ)

Proof of Theorem excomim

Step	Hyp	Ref	Expression
1		19.8a 712	. . . 4 ⊢ (φ → ∃xφ)
2	1	19.22i 723	. . 3 ⊢ (∃yφ → ∃y∃xφ)
3	2	19.22i 723	. 2 ⊢ (∃x∃yφ → ∃x∃y∃xφ)
4		hbe1 709	. . . 4 ⊢ (∃xφ → ∀x∃xφ)
5	4	hbex 701	. . 3 ⊢ (∃y∃xφ → ∀x∃y∃xφ)
6	5	19.9r 718	. 2 ⊢ (∃y∃xφ ↔ ∃x∃y∃xφ)
7	3, 6	sylibr 175	1 ⊢ (∃x∃yφ → ∃y∃xφ)

Syntax hints:   → wi 2  ∃wex 678

This theorem is referenced by:  excom 728  2euswap 1065  prnmadd 3894

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6ax-4 673  ax-5 674  ax-6 675<SPAN>  ax-7 676  ax-gen 677

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679

metamath.org