Theorem birexda 1214

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

Hypotheses

Ref	Expression
biralda.1	|- (ph -> A.xph)
biralda.2	|- ((ph /\ x e. A) -> (ps <-> ch))

Assertion

Ref	Expression
birexda	|- (ph -> (E.x e. A ps <-> E.x e. A ch))

Proof of Theorem birexda

Step	Hyp	Ref	Expression
1		biralda.1	. . 3 |- (ph -> A.xph)
2		biralda.2	. . . . 5 |- ((ph /\ x e. A) -> (ps <-> ch))
3	2	exp 291	. . . 4 |- (ph -> (x e. A -> (ps <-> ch)))
4	3	pm5.32d 491	. . 3 |- (ph -> ((x e. A /\ ps) <-> (x e. A /\ ch)))
5	1, 4	biexd 783	. 2 |- (ph -> (E.x(x e. A /\ ps) <-> E.x(x e. A /\ ch)))
6		df-rex 1206	. 2 |- (E.x e. A ps <-> E.x(x e. A /\ ps))
7		df-rex 1206	. 2 |- (E.x e. A ch <-> E.x(x e. A /\ ch))
8	5, 6, 7	3bitr4g 428	1 |- (ph -> (E.x e. A ps <-> E.x e. A ch))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672  E.wex 678   e. wcel 1092  E.wrex 1202

This theorem is referenced by:  birexdva 1216  birexd 1218

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-gen 677

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-rex 1206

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