Theorem reubiia 2650

Description: Formula-building rule for restricted existential quantifier (inference form). (Contributed by NM, 14-Nov-2004.)

Hypothesis

Ref	Expression
reubiia.1	|- ( x e. A -> ( ph <-> ps ) )

Assertion

Ref	Expression
reubiia	|- ( E! x e. A ph <-> E! x e. A ps )

Proof of Theorem reubiia

Step	Hyp	Ref	Expression
1		reubiia.1	. . . 4 |- ( x e. A -> ( ph <-> ps ) )
2	1	pm5.32i 450	. . 3 |- ( ( x e. A /\ ph ) <-> ( x e. A /\ ps ) )
3	2	eubii 2023	. 2 |- ( E! x ( x e. A /\ ph ) <-> E! x ( x e. A /\ ps ) )
4		df-reu 2451	. 2 |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
5		df-reu 2451	. 2 |- ( E! x e. A ps <-> E! x ( x e. A /\ ps ) )
6	3, 4, 5	3bitr4i 211	1 |- ( E! x e. A ph <-> E! x e. A ps )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   E!weu 2014   e. wcel 2136   E!wreu 2446

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 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-17 1514  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-eu 2017  df-reu 2451

This theorem is referenced by:  reubii  2651