Theorem bj-spimvwt 32656

Description: Closed form of spimvw 1927. See also spimt 2253. (Contributed by BJ, 8-Nov-2021.)

Assertion

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

Distinct variable groups:   x, y   ps, x

Allowed substitution hints:   ph( x, y)   ps( y)

Proof of Theorem bj-spimvwt

Step	Hyp	Ref	Expression
1		bj-alequexv 32655	. 2 |- ( A. x ( x = y -> ( ph -> ps ) ) -> E. x ( ph -> ps ) )
2		19.36v 1904	. 2 |- ( E. x ( ph -> ps ) <-> ( A. x ph -> ps ) )
3	1, 2	sylib 208	1 |- ( A. x ( x = y -> ( ph -> ps ) ) -> ( A. x ph -> ps ) )

Syntax hints:   -> 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  ax-5 1839  ax-6 1888

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

This theorem is referenced by: (None)