Theorem ax10oe 1678

Description: Quantifier Substitution for existential quantifiers. Analogue to ax10o 1603 but for E. rather than A.. (Contributed by Jim Kingdon, 21-Dec-2017.)

Assertion

Ref	Expression
ax10oe	|- ( A. x x = y -> ( E. x ps -> E. y ps ) )

Proof of Theorem ax10oe

Step	Hyp	Ref	Expression
1		ax-ia3 101	. . . 4 |- ( x = y -> ( ps -> ( x = y /\ ps ) ) )
2	1	alimi 1344	. . 3 |- ( A. x x = y -> A. x ( ps -> ( x = y /\ ps ) ) )
3		exim 1490	. . 3 |- ( A. x ( ps -> ( x = y /\ ps ) ) -> ( E. x ps -> E. x ( x = y /\ ps ) ) )
4	2, 3	syl 14	. 2 |- ( A. x x = y -> ( E. x ps -> E. x ( x = y /\ ps ) ) )
5		ax11e 1677	. . 3 |- ( x = y -> ( E. x ( x = y /\ ps ) -> E. y ps ) )
6	5	sps 1430	. 2 |- ( A. x x = y -> ( E. x ( x = y /\ ps ) -> E. y ps ) )
7	4, 6	syld 40	1 |- ( A. x x = y -> ( E. x ps -> E. y ps ) )

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-11 1397  ax-4 1400  ax-ial 1427

This theorem depends on definitions:  df-bi 110

This theorem is referenced by: (None)