Theorem reubiia 2538

Description: Formula-building rule for restricted existential quantifier (inference rule). (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 441	. . 3 |- ( ( x e. A /\ ph ) <-> ( x e. A /\ ps ) )
3	2	eubii 1950	. 2 |- ( E! x ( x e. A /\ ph ) <-> E! x ( x e. A /\ ps ) )
4		df-reu 2355	. 2 |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
5		df-reu 2355	. 2 |- ( E! x e. A ps <-> E! x ( x e. A /\ ps ) )
6	3, 4, 5	3bitr4i 210	1 |- ( E! x e. A ph <-> E! x e. A ps )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   e. wcel 1433   E!weu 1941   E!wreu 2350

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 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-eu 1944  df-reu 2355

This theorem is referenced by:  reubii  2539