Theorem rexbida 2630

Description: Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 6-Oct-2003.)

Hypotheses

Ref	Expression
ralbida.1	|- F/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 |- F/xph
2		ralbida.2	. . . 4 |- ((ph /\ x e. A) -> (ps <-> ch))
3	2	pm5.32da 622	. . 3 |- (ph -> ((x e. A /\ ps) <-> (x e. A /\ ch)))
4	1, 3	exbid 1773	. 2 |- (ph -> (E.x(x e. A /\ ps) <-> E.x(x e. A /\ ch)))
5		df-rex 2621	. 2 |- (E.x e. A ps <-> E.x(x e. A /\ ps))
6		df-rex 2621	. 2 |- (E.x e. A ch <-> E.x(x e. A /\ ch))
7	4, 5, 6	3bitr4g 279	1 |- (ph -> (E.x e. A ps <-> E.x e. A ch))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  E.wex 1541   F/wnf 1544   e. wcel 1710  E.wrex 2616

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-ex 1542  df-nf 1545  df-rex 2621

This theorem is referenced by:  rexbidva  2632  rexbid  2634  dfiun2g  4000  fun11iun  5306