Theorem rexbida 3047

Description: Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 6-Oct-2003.)

Hypotheses

Ref	Expression
rexbida.1	|- F/ x ph
rexbida.2	|- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )

Assertion

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

Proof of Theorem rexbida

Step	Hyp	Ref	Expression
1		rexbida.1	. . 3 |- F/ x ph
2		rexbida.2	. . . 4 |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )
3	2	pm5.32da 673	. . 3 |- ( ph -> ( ( x e. A /\ ps ) <-> ( x e. A /\ ch ) ) )
4	1, 3	exbid 2091	. 2 |- ( ph -> ( E. x ( x e. A /\ ps ) <-> E. x ( x e. A /\ ch ) ) )
5		df-rex 2918	. 2 |- ( E. x e. A ps <-> E. x ( x e. A /\ ps ) )
6		df-rex 2918	. 2 |- ( E. x e. A ch <-> E. x ( x e. A /\ ch ) )
7	4, 5, 6	3bitr4g 303	1 |- ( ph -> ( E. x e. A ps <-> E. x e. A ch ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   E.wex 1704   F/wnf 1708   e. wcel 1990   E.wrex 2913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-nf 1710  df-rex 2918

This theorem is referenced by:  rexbidvaALT  3050  rexbid  3051  dfiun2g  4552  fun11iun  7126  iuneq12daf  29373  bnj1366  30900  glbconxN  34664  supminfrnmpt  39672  limsupre2mpt  39962  limsupre3mpt  39966  limsupreuzmpt  39971