Theorem rcla42v 1404

Description: 2-variable restricted specialization with implicit substitution.

Hypotheses

Ref	Expression
rcla42v.1	|- (x = A -> (ph <-> ch))
rcla42v.2	|- (y = B -> (ch <-> ps))

Assertion

Ref	Expression
rcla42v	|- (A.x e. C A.y e. D ph -> ((A e. C /\ B e. D) -> ps))

Distinct variable group(s):   x,y,A   x,C   x,D   y,B   y,D   ch,x   ps,y

Proof of Theorem rcla42v

Step	Hyp	Ref	Expression
1		rcla42v.1	. . . 4 |- (x = A -> (ph <-> ch))
2	1	biraldv 1219	. . 3 |- (x = A -> (A.y e. D ph <-> A.y e. D ch))
3	2	rcla4v 1402	. 2 |- (A.x e. C A.y e. D ph -> (A e. C -> A.y e. D ch))
4		rcla42v.2	. . . . 5 |- (y = B -> (ch <-> ps))
5	4	rcla4v 1402	. . . 4 |- (A.y e. D ch -> (B e. D -> ps))
6	5	syl3 18	. . 3 |- ((A e. C -> A.y e. D ch) -> (A e. C -> (B e. D -> ps)))
7	6	imp3a 279	. 2 |- ((A e. C -> A.y e. D ch) -> ((A e. C /\ B e. D) -> ps))
8	3, 7	syl 12	1 |- (A.x e. C A.y e. D ph -> ((A e. C /\ B e. D) -> ps))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196   = wceq 1091   e. wcel 1092  A.wral 1201

This theorem is referenced by:  isorel 2932  isocnv 2934  isotr 2935  isotrALT 2936  fiint 3445  seqrn 4673  infxpidmlem7 4939  shaddclt 5123  shmulclt 5124  stjt 5676  stcltr1 5707

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-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-12 802  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349

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