Theorem speiv 1783

Description: Inference from existential specialization, using implicit substitition. (Contributed by NM, 19-Aug-1993.)

Hypotheses

Ref	Expression
speiv.1	|- ( x = y -> ( ph <-> ps ) )
speiv.2	|- ps

Assertion

Ref	Expression
speiv	|- E. x ph

Distinct variable group:   ps, x

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

Proof of Theorem speiv

Step	Hyp	Ref	Expression
1		speiv.2	. 2 |- ps
2		speiv.1	. . . 4 |- ( x = y -> ( ph <-> ps ) )
3	2	biimprd 156	. . 3 |- ( x = y -> ( ps -> ph ) )
4	3	spimev 1782	. 2 |- ( ps -> E. x ph )
5	1, 4	ax-mp 7	1 |- E. x ph

Syntax hints:   -> wi 4   <-> wb 103   E.wex 1421

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390

This theorem is referenced by: (None)