Theorem rcla4cva 1876

Description: Restricted specialization with implicit substitution.

Hypothesis

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

Assertion

Ref	Expression
rcla4cva	|- ((A.x e. B ph /\ A e. B) -> ps)

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

Proof of Theorem rcla4cva

Step	Hyp	Ref	Expression
1		rcla4v.1	. . 3 |- (x = A -> (ph <-> ps))
2	1	rcla4va 1875	. 2 |- ((A e. B /\ A.x e. B ph) -> ps)
3	2	ancoms 436	1 |- ((A.x e. B ph /\ A e. B) -> ps)

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

This theorem is referenced by:  disjne 2315  fconstfv 3849  odi 4210  omsmolem 4256  unblem1 4540  supmo 4576  sqr2irrlem3 6726  cau3ir 6915  climrecl 7110  climge0 7112  climcau 7156  infxpidmlem10 7561  elcls3 7711  iscncl 7770  metcnp 7887  cmscvg 7947  grpidinvlem3 8050  grpidinv 8052  grpidinv2 8060  vcid 8170  lnopeq0 9932  csmdsym 10261

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

Copyright terms: Public domain