Theorem reubidv 2796

Description: Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 17-Oct-1996.)

Hypothesis

Ref	Expression
reubidv.1	|- (ph -> (ps <-> ch))

Assertion

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

Distinct variable group:   ph,x

Allowed substitution hints:   ps(x)   ch(x)   A(x)

Proof of Theorem reubidv

Step	Hyp	Ref	Expression
1		reubidv.1	. . 3 |- (ph -> (ps <-> ch))
2	1	adantr 451	. 2 |- ((ph /\ x e. A) -> (ps <-> ch))
3	2	reubidva 2795	1 |- (ph -> (E!x e. A ps <-> E!x e. A ch))

Syntax hints:   -> wi 4   <-> wb 176   e. wcel 1710  E!wreu 2617

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-eu 2208  df-reu 2622

This theorem is referenced by:  reueqd  2818  sbcreug  3123