Theorem bj-ax12ig 32615

Description: A lemma used to prove a weak form of the axiom of substitution. A generalization of bj-ax12i 32616. (Contributed by BJ, 19-Dec-2020.)

Hypotheses

Ref	Expression
bj-ax12ig.1	|- ( ph -> ( ps <-> ch ) )
bj-ax12ig.2	|- ( ph -> ( ch -> A. x ch ) )

Assertion

Ref	Expression
bj-ax12ig	|- ( ph -> ( ps -> A. x ( ph -> ps ) ) )

Proof of Theorem bj-ax12ig

Step	Hyp	Ref	Expression
1		bj-ax12ig.1	. . . 4 |- ( ph -> ( ps <-> ch ) )
2	1	pm5.32i 669	. . 3 |- ( ( ph /\ ps ) <-> ( ph /\ ch ) )
3		bj-ax12ig.2	. . . . 5 |- ( ph -> ( ch -> A. x ch ) )
4	3	imp 445	. . . 4 |- ( ( ph /\ ch ) -> A. x ch )
5	1	biimprcd 240	. . . 4 |- ( ch -> ( ph -> ps ) )
6	4, 5	sylg 1750	. . 3 |- ( ( ph /\ ch ) -> A. x ( ph -> ps ) )
7	2, 6	sylbi 207	. 2 |- ( ( ph /\ ps ) -> A. x ( ph -> ps ) )
8	7	ex 450	1 |- ( ph -> ( ps -> A. x ( ph -> ps ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481

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-an 386

This theorem is referenced by:  bj-ax12i  32616