Theorem rcla4va 1875

Description: Restricted specialization with implicit substitution.

Hypothesis

Ref	Expression
rcla4v.1	|- (x = A -> (ph <-> ps))

Assertion

Ref	Expression
rcla4va	|- ((A e. B /\ A.x e. B ph) -> ps)

Distinct variable groups:   x,A   x,B   ps,x

Proof of Theorem rcla4va

Step	Hyp	Ref	Expression
1		rcla4v.1	. . 3 |- (x = A -> (ph <-> ps))
2	1	rcla4v 1873	. 2 |- (A e. B -> (A.x e. B ph -> ps))
3	2	imp 350	1 |- ((A e. B /\ A.x e. B ph) -> ps)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  A.wral 1645

This theorem is referenced by:  rcla4cva 1876  unifiOLD 4557  supmo 4576  noinfep 4640  nnleltp1t 5954  infmrcl 6069  xrsupsslem 6076  xrinfmsslem 6077  supxrunb1 6089  supxrunb2 6090  zextlet 6189  fsequb 6523  seqzfveq 6554  faclbnd4lem4 6951  climaddlem1 7114  climmullem6 7125  serzf0 7169  ser1f0 7170  isum1p 7206  iserzgt0 7211  opnin 7869  lpbl 7880  bcthlem18 8016  grpidinvlem3 8050  grpideu 8053  lnopcon 9963  lnfncon 9990  nlelch 9994  homcard 10539

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-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-v 1812

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