Theorem spime 2256

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 11	. . 3 |- ( T. -> F/ x ph )
3		spime.2	. . 3 |- ( x = y -> ( ph -> ps ) )
4	2, 3	spimed 2255	. 2 |- ( T. -> ( ph -> E. x ps ) )
5	4	trud 1493	1 |- ( ph -> E. x ps )

Syntax hints:   -> wi 4   T. wtru 1484   E.wex 1704   F/wnf 1708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710

This theorem is referenced by:  spimev  2259  exnel  31708