Theorem spimed 1754

Description: Deduction version of spime 1755. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 19-Feb-2018.)

Hypotheses

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

Assertion

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

Proof of Theorem spimed

Step	Hyp	Ref	Expression
1		spimed.1	. . 3 |- ( ch -> F/ x ph )
2	1	nfrd 1534	. 2 |- ( ch -> ( ph -> A. x ph ) )
3		a9e 1710	. . . 4 |- E. x x = y
4		spimed.2	. . . 4 |- ( x = y -> ( ph -> ps ) )
5	3, 4	eximii 1616	. . 3 |- E. x ( ph -> ps )
6	5	19.35i 1639	. 2 |- ( A. x ph -> E. x ps )
7	2, 6	syl6 33	1 |- ( ch -> ( ph -> E. x ps ) )

Syntax hints:   -> wi 4   A.wal 1362   F/wnf 1474   E.wex 1506

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-i9 1544  ax-ial 1548

This theorem depends on definitions:  df-bi 117  df-nf 1475

This theorem is referenced by:  spime  1755