Theorem rexbid 2437

Description: Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 27-Jun-1998.)

Hypotheses

Ref	Expression
ralbid.1	|- F/ x ph
ralbid.2	|- ( ph -> ( ps <-> ch ) )

Assertion

Ref	Expression
rexbid	|- ( ph -> ( E. x e. A ps <-> E. x e. A ch ) )

Proof of Theorem rexbid

Step	Hyp	Ref	Expression
1		ralbid.1	. 2 |- F/ x ph
2		ralbid.2	. . 3 |- ( ph -> ( ps <-> ch ) )
3	2	adantr 274	. 2 |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )
4	1, 3	rexbida 2433	1 |- ( ph -> ( E. x e. A ps <-> E. x e. A ch ) )

Syntax hints:   -> wi 4   <-> wb 104   F/wnf 1437   e. wcel 1481   E.wrex 2418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-ial 1515

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-rex 2423

This theorem is referenced by:  rexbidv  2439  sbcrext  2990  mkvprop  7040  caucvgsrlemgt1  7627  bezout  11735