Theorem exbid 1507

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 1412	. 2 |- ( ph -> A. x ph )
3		exbid.2	. 2 |- ( ph -> ( ps <-> ch ) )
4	2, 3	exbidh 1505	1 |- ( ph -> ( E. x ps <-> E. x ch ) )

Syntax hints:   -> wi 4   <-> wb 98   F/wnf 1349   E.wex 1381

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 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-nf 1350

This theorem is referenced by:  mobid  1935  rexbida  2321  rexeqf  2502  opabbid  3822  repizf2  3915  oprabbid  5558  sscoll2  10113