Theorem bj-ax12w 32665

Description: The general statement that ax12w 2010 proves. (Contributed by BJ, 20-Mar-2020.)

Hypotheses

Ref	Expression
bj-ax12w.1	|- ( ph -> ( ps <-> ch ) )
bj-ax12w.2	|- ( y = z -> ( ps <-> th ) )

Assertion

Ref	Expression
bj-ax12w	|- ( ph -> ( A. y ps -> A. x ( ph -> ps ) ) )

Distinct variable groups:   ch, x   th, y   ps, z   y, z

Allowed substitution hints:   ph( x, y, z)   ps( x, y)   ch( y, z)   th( x, z)

Proof of Theorem bj-ax12w

Step	Hyp	Ref	Expression
1		bj-ax12w.2	. . 3 |- ( y = z -> ( ps <-> th ) )
2	1	spw 1967	. 2 |- ( A. y ps -> ps )
3		bj-ax12w.1	. . 3 |- ( ph -> ( ps <-> ch ) )
4	3	bj-ax12wlem 32617	. 2 |- ( ph -> ( ps -> A. x ( ph -> ps ) ) )
5	2, 4	syl5 34	1 |- ( ph -> ( A. y ps -> A. x ( ph -> ps ) ) )

Syntax hints:   -> wi 4   <-> wb 196   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  ax-5 1839  ax-6 1888  ax-7 1935

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

This theorem is referenced by: (None)