Theorem a4c1 844

Description: A more general version of a4c 843.

Hypotheses

Ref	Expression
a4c1.1	|- (ch -> A.xch)
a4c1.2	|- (ch -> (ph -> A.xph))
a4c1.3	|- (x = y -> (ph -> ps))

Assertion

Ref	Expression
a4c1	|- (ch -> (ph -> E.xps))

Proof of Theorem a4c1

Step	Hyp	Ref	Expression
1		a4c1.1	. . . . . 6 |- (ch -> A.xch)
2	1	adantr 306	. . . . 5 |- ((ch /\ ph) -> A.xch)
3		a4c1.2	. . . . . 6 |- (ch -> (ph -> A.xph))
4	3	imp 277	. . . . 5 |- ((ch /\ ph) -> A.xph)
5	2, 4	jca 236	. . . 4 |- ((ch /\ ph) -> (A.xch /\ A.xph))
6		19.26 749	. . . 4 |- (A.x(ch /\ ph) <-> (A.xch /\ A.xph))
7	5, 6	sylibr 175	. . 3 |- ((ch /\ ph) -> A.x(ch /\ ph))
8		a4c1.3	. . . 4 |- (x = y -> (ph -> ps))
9	8	adantld 307	. . 3 |- (x = y -> ((ch /\ ph) -> ps))
10	7, 9	a4c 843	. 2 |- ((ch /\ ph) -> E.xps)
11	10	exp 291	1 |- (ch -> (ph -> E.xps))

Syntax hints:   -> wi 2   /\ wa 196  A.wal 672  E.wex 678   = weq 797

This theorem is referenced by:  eqvin.l1 851

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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