Theorem bj-ceqsalg0 32877

Description: The FOL content of ceqsalg 3230. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-ceqsalg0.1	|- F/ x ps
bj-ceqsalg0.2	|- ( ch -> ( ph <-> ps ) )

Assertion

Ref	Expression
bj-ceqsalg0	|- ( E. x ch -> ( A. x ( ch -> ph ) <-> ps ) )

Proof of Theorem bj-ceqsalg0

Step	Hyp	Ref	Expression
1		bj-ceqsalg0.1	. 2 |- F/ x ps
2		bj-ceqsalg0.2	. . 3 |- ( ch -> ( ph <-> ps ) )
3	2	ax-gen 1722	. 2 |- A. x ( ch -> ( ph <-> ps ) )
4		bj-ceqsalt0 32873	. 2 |- ( ( F/ x ps /\ A. x ( ch -> ( ph <-> ps ) ) /\ E. x ch ) -> ( A. x ( ch -> ph ) <-> ps ) )
5	1, 3, 4	mp3an12 1414	1 |- ( E. x ch -> ( A. x ( ch -> ph ) <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 196   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-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-ex 1705  df-nf 1710

This theorem is referenced by:  bj-ceqsalg  32878  bj-ceqsalgv  32880