Theorem ceqsexv 2587

Description: Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.)

Hypotheses

Ref	Expression
ceqsexv.1	|- A e. _V
ceqsexv.2	|- (x = A -> (ph <-> ps))

Assertion

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

Distinct variable groups:   x,A   ps,x

Allowed substitution hint:   ph(x)

Proof of Theorem ceqsexv

Step	Hyp	Ref	Expression
1		nfv 1418	. 2 |- F/xps
2		ceqsexv.1	. 2 |- A e. _V
3		ceqsexv.2	. 2 |- (x = A -> (ph <-> ps))
4	1, 2, 3	ceqsex 2586	1 |- (E.x(x = A /\ ph) <-> ps)

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1242  E.wex 1378   e. wcel 1390   _Vcvv 2551

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-v 2553

This theorem is referenced by:  ceqsex3v  2590  gencbvex  2594  sbhypf  2597  euxfr2dc  2720  inuni  3900  eqvinop  3971  onm  4104  uniuni  4149  opeliunxp  4338  elvvv  4346  rexiunxp  4421  imai  4624  coi1  4779  abrexco  5341  opabex3d  5690  opabex3  5691  xpsnen  6231  xpcomco  6236  xpassen  6240