Theorem rcla4dv 1869

Description: Restricted specialization with implicit substitution.

Hypothesis

Ref	Expression
rcla4dv.1	|- ((ph /\ x = A) -> (ps <-> ch))

Assertion

Ref	Expression
rcla4dv	|- ((ph /\ A e. B) -> (A.x e. B ps -> ch))

Distinct variable groups:   x,A   x,B   ph,x   ch,x

Proof of Theorem rcla4dv

Step	Hyp	Ref	Expression
1		rcla4dv.1	. . . . . . 7 |- ((ph /\ x = A) -> (ps <-> ch))
2	1	expcom 374	. . . . . 6 |- (x = A -> (ph -> (ps <-> ch)))
3	2	pm5.74d 583	. . . . 5 |- (x = A -> ((ph -> ps) <-> (ph -> ch)))
4	3	rcla4v 1864	. . . 4 |- (A e. B -> (A.x e. B (ph -> ps) -> (ph -> ch)))
5		r19.21v 1708	. . . 4 |- (A.x e. B (ph -> ps) <-> (ph -> A.x e. B ps))
6	4, 5	syl5ibr 207	. . 3 |- (A e. B -> ((ph -> A.x e. B ps) -> (ph -> ch)))
7	6	pm2.86d 71	. 2 |- (A e. B -> (ph -> (A.x e. B ps -> ch)))
8	7	impcom 351	1 |- ((ph /\ A e. B) -> (A.x e. B ps -> ch))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  A.wral 1637

This theorem is referenced by:  ralxfrd 2887  imonclem 10583

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-ral 1641  df-v 1803

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