Theorem bireua 1319

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

Hypothesis

Ref	Expression
bireua.1	|- (x e. A -> (ph <-> ps))

Assertion

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

Proof of Theorem bireua

Step	Hyp	Ref	Expression
1		bireua.1	. . . 4 |- (x e. A -> (ph <-> ps))
2	1	pm5.32i 489	. . 3 |- ((x e. A /\ ph) <-> (x e. A /\ ps))
3	2	bieu 1014	. 2 |- (E!x(x e. A /\ ph) <-> E!x(x e. A /\ ps))
4		df-reu 1207	. 2 |- (E!x e. A ph <-> E!x(x e. A /\ ph))
5		df-reu 1207	. 2 |- (E!x e. A ps <-> E!x(x e. A /\ ps))
6	3, 4, 5	3bitr4 158	1 |- (E!x e. A ph <-> E!x e. A ps)

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  E!weu 1007   e. wcel 1092  E!wreu 1203

This theorem is referenced by:  bireu 1320  reuxfr2 1579  reuxfr 1580

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677  ax-9 799  ax-12 802  ax-17 925

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-eu 1009  df-reu 1207

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