Theorem spimeh 1667

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.xph)
spimeh.2	|- (x = z -> (ph -> ps))

Assertion

Ref	Expression
spimeh	|- (ph -> E.xps)

Distinct variable group:   x,z

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

Proof of Theorem spimeh

Step	Hyp	Ref	Expression
1		spimeh.1	. 2 |- (ph -> A.xph)
2		a9ev 1656	. . . 4 |- E.x x = z
3		spimeh.2	. . . . 5 |- (x = z -> (ph -> ps))
4	3	eximi 1576	. . . 4 |- (E.x x = z -> E.x(ph -> ps))
5	2, 4	ax-mp 5	. . 3 |- E.x(ph -> ps)
6	5	19.35i 1601	. 2 |- (A.xph -> E.xps)
7	1, 6	syl 15	1 |- (ph -> E.xps)

Syntax hints:   -> wi 4  A.wal 1540  E.wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-9 1654

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542

This theorem is referenced by:  ax12olem1  1927  ax10lem2  1937