Theorem reubiia 3127

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 669	. . 3 |- ( ( x e. A /\ ph ) <-> ( x e. A /\ ps ) )
3	2	eubii 2492	. 2 |- ( E! x ( x e. A /\ ph ) <-> E! x ( x e. A /\ ps ) )
4		df-reu 2919	. 2 |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
5		df-reu 2919	. 2 |- ( E! x e. A ps <-> E! x ( x e. A /\ ps ) )
6	3, 4, 5	3bitr4i 292	1 |- ( E! x e. A ph <-> E! x e. A ps )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   E!weu 2470   E!wreu 2914

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-tru 1486  df-ex 1705  df-nf 1710  df-eu 2474  df-reu 2919

This theorem is referenced by:  reubii  3128  riotaxfrd  6642  infempty  8412