Theorem rexbida 1650

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

Hypotheses

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

Assertion

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

Proof of Theorem rexbida

Step	Hyp	Ref	Expression
1		ralbida.1	. . 3 |- (ph -> A.xph)
2		ralbida.2	. . . 4 |- ((ph /\ x e. A) -> (ps <-> ch))
3	2	pm5.32da 647	. . 3 |- (ph -> ((x e. A /\ ps) <-> (x e. A /\ ch)))
4	1, 3	exbid 1101	. 2 |- (ph -> (E.x(x e. A /\ ps) <-> E.x(x e. A /\ ch)))
5		df-rex 1642	. 2 |- (E.x e. A ps <-> E.x(x e. A /\ ps))
6		df-rex 1642	. 2 |- (E.x e. A ch <-> E.x(x e. A /\ ch))
7	4, 5, 6	3bitr4g 553	1 |- (ph -> (E.x e. A ps <-> E.x e. A ch))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 951   e. wcel 955  E.wex 977  E.wrex 1638

This theorem is referenced by:  rexbidva 1652  rexbid 1654  elrnopabg 3785  elrnoprabg 4108

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 960  ax-4 970  ax-5o 972

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-rex 1642

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