Theorem ax9o 1588

Description: An implication related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)

Assertion

Ref	Expression
ax9o	|- ( A. x ( x = y -> A. x ph ) -> ph )

Proof of Theorem ax9o

Step	Hyp	Ref	Expression
1		a9e 1586	. 2 |- E. x x = y
2		19.29r 1512	. . 3 |- ( ( E. x x = y /\ A. x ( x = y -> A. x ph ) ) -> E. x ( x = y /\ ( x = y -> A. x ph ) ) )
3		hba1 1433	. . . . 5 |- ( A. x ph -> A. x A. x ph )
4		pm3.35 329	. . . . 5 |- ( ( x = y /\ ( x = y -> A. x ph ) ) -> A. x ph )
5	3, 4	exlimih 1484	. . . 4 |- ( E. x ( x = y /\ ( x = y -> A. x ph ) ) -> A. x ph )
6		ax-4 1400	. . . 4 |- ( A. x ph -> ph )
7	5, 6	syl 14	. . 3 |- ( E. x ( x = y /\ ( x = y -> A. x ph ) ) -> ph )
8	2, 7	syl 14	. 2 |- ( ( E. x x = y /\ A. x ( x = y -> A. x ph ) ) -> ph )
9	1, 8	mpan 400	1 |- ( A. x ( x = y -> A. x ph ) -> ph )

Syntax hints:   -> wi 4   /\ wa 97   A.wal 1241   = wceq 1243   E.wex 1381

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-i9 1423  ax-ial 1427

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  equsalh  1614  spimth  1623  spimh  1625