Theorem ceqsexgv 1884

Description: Elimination of an existential quantifier, using implicit substitution.

Hypothesis

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

Assertion

Ref	Expression
ceqsexgv	|- (A e. B -> (E.x(x = A /\ ph) <-> ps))

Distinct variable groups:   x,A   ps,x

Proof of Theorem ceqsexgv

Step	Hyp	Ref	Expression
1		ax-17 969	. 2 |- (ps -> A.xps)
2		ceqsexgv.1	. 2 |- (x = A -> (ph <-> ps))
3	1, 2	ceqsexg 1883	1 |- (A e. B -> (E.x(x = A /\ ph) <-> ps))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 954   e. wcel 956  E.wex 978

This theorem is referenced by:  ceqsrexv 1885  clel3g 1888  imasng 3416  elxp5 3446  fvopabn 3777  2ndconst 4087  xpsnen 4421  ismet 7748  isgrp 7991  spwval2 8595  bsi 10418

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808

Copyright terms: Public domain