Theorem reubii 2539

Description: Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 22-Oct-1999.)

Hypothesis

Ref	Expression
reubii.1	|- ( ph <-> ps )

Assertion

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

Proof of Theorem reubii

Step	Hyp	Ref	Expression
1		reubii.1	. . 3 |- ( ph <-> ps )
2	1	a1i 9	. 2 |- ( x e. A -> ( ph <-> ps ) )
3	2	reubiia 2538	1 |- ( E! x e. A ph <-> E! x e. A ps )

Syntax hints:   <-> wb 103   e. wcel 1433   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:  caucvgsrlemcl  6965  axcaucvglemcl  7061  axcaucvglemval  7063