Theorem bj-alcomexcom 32670

Description: Commutation of universal quantifiers implies commutation of existential quantifiers. Can be placed in the ax-4 1737 section, soon after 2nexaln 1757, and used to prove excom 2042. (Contributed by BJ, 29-Nov-2020.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-alcomexcom	|- ( ( A. x A. y -. ph -> A. y A. x -. ph ) -> ( E. y E. x ph -> E. x E. y ph ) )

Proof of Theorem bj-alcomexcom

Step	Hyp	Ref	Expression
1		2nexaln 1757	. . 3 |- ( -. E. x E. y ph <-> A. x A. y -. ph )
2		2nexaln 1757	. . 3 |- ( -. E. y E. x ph <-> A. y A. x -. ph )
3	1, 2	imbi12i 340	. 2 |- ( ( -. E. x E. y ph -> -. E. y E. x ph ) <-> ( A. x A. y -. ph -> A. y A. x -. ph ) )
4		con4 112	. 2 |- ( ( -. E. x E. y ph -> -. E. y E. x ph ) -> ( E. y E. x ph -> E. x E. y ph ) )
5	3, 4	sylbir 225	1 |- ( ( A. x A. y -. ph -> A. y A. x -. ph ) -> ( E. y E. x ph -> E. x E. y ph ) )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1481   E.wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737

This theorem depends on definitions:  df-bi 197  df-ex 1705

This theorem is referenced by: (None)