Theorem spimeh 1739

Description: Existential introduction, using implicit substitition. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Revised by NM, 3-Feb-2015.) (New usage is discouraged.)

Hypotheses

Ref	Expression
spimeh.1	|- ( ph -> A. x ph )
spimeh.2	|- ( x = y -> ( ph -> ps ) )

Assertion

Ref	Expression
spimeh	|- ( ph -> E. x ps )

Proof of Theorem spimeh

Step	Hyp	Ref	Expression
1		a9e 1696	. 2 |- E. x x = y
2		spimeh.1	. . 3 |- ( ph -> A. x ph )
3		spimeh.2	. . . 4 |- ( x = y -> ( ph -> ps ) )
4	3	com12 30	. . 3 |- ( ph -> ( x = y -> ps ) )
5	2, 4	eximdh 1611	. 2 |- ( ph -> ( E. x x = y -> E. x ps ) )
6	1, 5	mpi 15	1 |- ( ph -> E. x ps )

Syntax hints:   -> wi 4   A.wal 1351   = wceq 1353   E.wex 1492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-i9 1530  ax-ial 1534

This theorem depends on definitions:  df-bi 117

This theorem is referenced by: (None)