Theorem spime 1669

Description: Existential introduction, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 6-Mar-2018.)

Hypotheses

Ref	Expression
spime.1	|- F/ x ph
spime.2	|- ( x = y -> ( ph -> ps ) )

Assertion

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

Proof of Theorem spime

Step	Hyp	Ref	Expression
1		spime.1	. . . 4 |- F/ x ph
2	1	a1i 9	. . 3 |- ( T. -> F/ x ph )
3		spime.2	. . 3 |- ( x = y -> ( ph -> ps ) )
4	2, 3	spimed 1668	. 2 |- ( T. -> ( ph -> E. x ps ) )
5	4	trud 1293	1 |- ( ph -> E. x ps )

Syntax hints:   -> wi 4   T. wtru 1285   F/wnf 1389   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-i9 1463  ax-ial 1467

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

This theorem is referenced by:  spimev  1782