Theorem rcla42ev 1881

Description: 2-variable restricted existential specialization with implicit substitution.

Hypotheses

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

Assertion

Ref	Expression
rcla42ev	|- ((A e. C /\ B e. D /\ ps) -> E.x e. C E.y e. D ph)

Distinct variable groups:   x,y,A   x,C   x,D   y,B   y,D   ch,x   ps,y

Proof of Theorem rcla42ev

Step	Hyp	Ref	Expression
1		rcla42v.2	. . . . 5 |- (y = B -> (ch <-> ps))
2	1	rcla4ev 1877	. . . 4 |- ((B e. D /\ ps) -> E.y e. D ch)
3	2	anim2i 335	. . 3 |- ((A e. C /\ (B e. D /\ ps)) -> (A e. C /\ E.y e. D ch))
4	3	3impb 829	. 2 |- ((A e. C /\ B e. D /\ ps) -> (A e. C /\ E.y e. D ch))
5		rcla42v.1	. . . 4 |- (x = A -> (ph <-> ch))
6	5	rexbidv 1664	. . 3 |- (x = A -> (E.y e. D ph <-> E.y e. D ch))
7	6	rcla4ev 1877	. 2 |- ((A e. C /\ E.y e. D ch) -> E.x e. C E.y e. D ph)
8	4, 7	syl 10	1 |- ((A e. C /\ B e. D /\ ps) -> E.x e. C E.y e. D ph)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  E.wrex 1646

This theorem is referenced by:  rcla4eopr 3990  2dom 4427  unxpdomlem 4843  quoremOLD 6252  retopbas 7655  blelrn 7848  methausi 7881  blssioo 7913  dtt2 10618

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-rex 1650  df-v 1812

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