Theorem reubiia 2797

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 618	. . 3 |- ((x e. A /\ ph) <-> (x e. A /\ ps))
3	2	eubii 2213	. 2 |- (E!x(x e. A /\ ph) <-> E!x(x e. A /\ ps))
4		df-reu 2622	. 2 |- (E!x e. A ph <-> E!x(x e. A /\ ph))
5		df-reu 2622	. 2 |- (E!x e. A ps <-> E!x(x e. A /\ ps))
6	3, 4, 5	3bitr4i 268	1 |- (E!x e. A ph <-> E!x e. A ps)

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   e. wcel 1710  E!weu 2204  E!wreu 2617

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-eu 2208  df-reu 2622

This theorem is referenced by:  reubii  2798