Theorem a4c 843

Description: Existential introduction with implicit substitution. Compare Lemma 14 of [Tarski] p. 70.

Hypotheses

Ref	Expression
a4c.1	|- (ph -> A.xph)
a4c.2	|- (x = y -> (ph -> ps))

Assertion

Ref	Expression
a4c	|- (ph -> E.xps)

Proof of Theorem a4c

Step	Hyp	Ref	Expression
1		a4c.1	. . . . 5 |- (ph -> A.xph)
2	1	hbne 699	. . . 4 |- (-. ph -> A.x -. ph)
3		a4c.2	. . . . 5 |- (x = y -> (ph -> ps))
4	3	con3d 87	. . . 4 |- (x = y -> (-. ps -> -. ph))
5	2, 4	a4a 842	. . 3 |- (A.x -. ps -> -. ph)
6	5	con2i 89	. 2 |- (ph -> -. A.x -. ps)
7		df-ex 679	. 2 |- (E.xps <-> -. A.x -. ps)
8	6, 7	sylibr 175	1 |- (ph -> E.xps)

Syntax hints:  -. wn 1   -> wi 2  A.wal 672  E.wex 678   = weq 797

This theorem is referenced by:  a4c1 844  a4w 929

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677  ax-9 799

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679

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