Theorem 2rexbidv 2349

Description: Formula-building rule for restricted existential quantifiers (deduction rule). (Contributed by NM, 28-Jan-2006.)

Hypothesis

Ref	Expression
2ralbidv.1	|- ( ph -> ( ps <-> ch ) )

Assertion

Ref	Expression
2rexbidv	|- ( ph -> ( E. x e. A E. y e. B ps <-> E. x e. A E. y e. B ch ) )

Distinct variable groups:   ph, x   ph, y

Allowed substitution hints:   ps( x, y)   ch( x, y)   A( x, y)   B( x, y)

Proof of Theorem 2rexbidv

Step	Hyp	Ref	Expression
1		2ralbidv.1	. . 3 |- ( ph -> ( ps <-> ch ) )
2	1	rexbidv 2327	. 2 |- ( ph -> ( E. y e. B ps <-> E. y e. B ch ) )
3	2	rexbidv 2327	1 |- ( ph -> ( E. x e. A E. y e. B ps <-> E. x e. A E. y e. B ch ) )

Syntax hints:   -> wi 4   <-> wb 98   E.wrex 2307

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-17 1419  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-rex 2312

This theorem is referenced by:  f1oiso  5465  elrnmpt2g  5613  elrnmpt2  5614  ralrnmpt2  5615  rexrnmpt2  5616  ovelrn  5649  eroveu  6197  genipv  6607  genpelxp  6609  genpelvl  6610  genpelvu  6611  axcnre  6955  apreap  7578  apreim  7594