Theorem ceqsexv 2843

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 1577	. 2 |- F/ x ps
2		ceqsexv.1	. 2 |- A e. _V
3		ceqsexv.2	. 2 |- ( x = A -> ( ph <-> ps ) )
4	1, 2, 3	ceqsex 2842	1 |- ( E. x ( x = A /\ ph ) <-> ps )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   E.wex 1541   e. wcel 2202   _Vcvv 2803

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-v 2805

This theorem is referenced by:  ceqsex3v  2847  gencbvex  2851  sbhypf  2854  euxfr2dc  2992  inuni  4250  eqvinop  4341  onm  4504  uniuni  4554  opeliunxp  4787  elvvv  4795  rexiunxp  4878  imai  5099  coi1  5259  abrexco  5910  opabex3d  6292  opabex3  6293  mapsnen  7029  xpsnen  7048  xpcomco  7053  xpassen  7057