Theorem exbid 1548

Description: Formula-building rule for existential quantifier (deduction rule). (Contributed by Mario Carneiro, 24-Sep-2016.)

Hypotheses

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

Assertion

Ref	Expression
exbid	|- ( ph -> ( E. x ps <-> E. x ch ) )

Proof of Theorem exbid

Step	Hyp	Ref	Expression
1		exbid.1	. . 3 |- F/ x ph
2	1	nfri 1453	. 2 |- ( ph -> A. x ph )
3		exbid.2	. 2 |- ( ph -> ( ps <-> ch ) )
4	2, 3	exbidh 1546	1 |- ( ph -> ( E. x ps <-> E. x ch ) )

Syntax hints:   -> wi 4   <-> wb 103   F/wnf 1390   E.wex 1422

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-ial 1468

This theorem depends on definitions:  df-bi 115  df-nf 1391

This theorem is referenced by:  mobid  1977  rexbida  2364  rexeqf  2547  opabbid  3851  repizf2  3944  oprabbid  5589  sscoll2  10968