Theorem spimeh 1925

Description: Existential introduction, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Proof shortened by Wolf Lammen, 10-Dec-2017.)

Hypotheses

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

Assertion

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

Distinct variable group:   x, y

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

Proof of Theorem spimeh

Step	Hyp	Ref	Expression
1		spimeh.1	. 2 |- ( ph -> A. x ph )
2		ax6ev 1890	. . . 4 |- E. x x = y
3		spimeh.2	. . . 4 |- ( x = y -> ( ph -> ps ) )
4	2, 3	eximii 1764	. . 3 |- E. x ( ph -> ps )
5	4	19.35i 1806	. 2 |- ( A. x ph -> E. x ps )
6	1, 5	syl 17	1 |- ( ph -> E. x ps )

Syntax hints:   -> wi 4   A.wal 1481   E.wex 1704

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-6 1888

This theorem depends on definitions:  df-bi 197  df-ex 1705

This theorem is referenced by:  bj-spimevw  32657  bj-cbvexiw  32659