Theorem reubiia 1784

Description: Formula-building rule for restricted existential quantifier (inference rule).

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 647	. . 3 |- ((x e. A /\ ph) <-> (x e. A /\ ps))
3	2	eubii 1389	. 2 |- (E!x(x e. A /\ ph) <-> E!x(x e. A /\ ps))
4		df-reu 1654	. 2 |- (E!x e. A ph <-> E!x(x e. A /\ ph))
5		df-reu 1654	. 2 |- (E!x e. A ps <-> E!x(x e. A /\ ps))
6	3, 4, 5	3bitr4 183	1 |- (E!x e. A ph <-> E!x e. A ps)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   e. wcel 960  E!weu 1382  E!wreu 1650

This theorem is referenced by:  reubii 1785  reuxfr2 2909  reuxfr 2910  reuunixfr 2912  pjtheu2 9245

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 965  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-eu 1384  df-reu 1654

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