Theorem bj-spimt2 32709

Description: A step in the proof of spimt 2253. (Contributed by BJ, 2-May-2019.)

Assertion

Ref	Expression
bj-spimt2	|- ( A. x ( x = y -> ( ph -> ps ) ) -> ( ( E. x ps -> ps ) -> ( A. x ph -> ps ) ) )

Proof of Theorem bj-spimt2

Step	Hyp	Ref	Expression
1		bj-alequex 32708	. . 3 |- ( A. x ( x = y -> ( ph -> ps ) ) -> E. x ( ph -> ps ) )
2		19.35 1805	. . 3 |- ( E. x ( ph -> ps ) <-> ( A. x ph -> E. x ps ) )
3	1, 2	sylib 208	. 2 |- ( A. x ( x = y -> ( ph -> ps ) ) -> ( A. x ph -> E. x ps ) )
4	3	imim1d 82	1 |- ( A. x ( x = y -> ( ph -> ps ) ) -> ( ( E. x ps -> 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  ax-7 1935  ax-12 2047  ax-13 2246

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

This theorem is referenced by:  bj-cbv3ta  32710