Theorem bj-cbv3tb 32711

Description: Closed form of cbv3 2265. (Contributed by BJ, 2-May-2019.)

Assertion

Ref	Expression
bj-cbv3tb	|- ( A. x A. y ( x = y -> ( ph -> ps ) ) -> ( ( A. y F/ x ps /\ A. x F/ y ph ) -> ( A. x ph -> A. y ps ) ) )

Proof of Theorem bj-cbv3tb

Step	Hyp	Ref	Expression
1		19.9t 2071	. . . 4 |- ( F/ x ps -> ( E. x ps <-> ps ) )
2	1	biimpd 219	. . 3 |- ( F/ x ps -> ( E. x ps -> ps ) )
3	2	alimi 1739	. 2 |- ( A. y F/ x ps -> A. y ( E. x ps -> ps ) )
4		nf5r 2064	. . 3 |- ( F/ y ph -> ( ph -> A. y ph ) )
5	4	alimi 1739	. 2 |- ( A. x F/ y ph -> A. x ( ph -> A. y ph ) )
6		bj-cbv3ta 32710	. 2 |- ( A. x A. y ( x = y -> ( ph -> ps ) ) -> ( ( A. y ( E. x ps -> ps ) /\ A. x ( ph -> A. y ph ) ) -> ( A. x ph -> A. y ps ) ) )
7	3, 5, 6	syl2ani 688	1 |- ( A. x A. y ( x = y -> ( ph -> ps ) ) -> ( ( A. y F/ x ps /\ A. x F/ y ph ) -> ( A. x ph -> A. y ps ) ) )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   E.wex 1704   F/wnf 1708

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-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-nf 1710

This theorem is referenced by: (None)