Theorem spime 1976

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.)

Hypotheses

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

Assertion

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

Proof of Theorem spime

Step	Hyp	Ref	Expression
1		spime.1	. . . . 5 |- F/xph
2	1	nfn 1793	. . . 4 |- F/x -. ph
3		spime.2	. . . . 5 |- (x = y -> (ph -> ps))
4	3	con3d 125	. . . 4 |- (x = y -> (-. ps -> -. ph))
5	2, 4	spim 1975	. . 3 |- (A.x -. ps -> -. ph)
6	5	con2i 112	. 2 |- (ph -> -. A.x -. ps)
7		df-ex 1542	. 2 |- (E.xps <-> -. A.x -. ps)
8	6, 7	sylibr 203	1 |- (ph -> E.xps)

Syntax hints:  -. wn 3   -> wi 4  A.wal 1540  E.wex 1541   F/wnf 1544

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by:  spimed  1977  spimev  1999