Theorem rabbidva 2850

Description: Equivalent wff's yield equal restricted class abstractions (deduction rule). (Contributed by NM, 28-Nov-2003.)

Hypothesis

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

Assertion

Ref	Expression
rabbidva	|- (ph -> {x e. A | ps} = {x e. A | ch})

Distinct variable group:   ph,x

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

Proof of Theorem rabbidva

Step	Hyp	Ref	Expression
1		rabbidva.1	. . 3 |- ((ph /\ x e. A) -> (ps <-> ch))
2	1	ralrimiva 2697	. 2 |- (ph -> A.x e. A (ps <-> ch))
3		rabbi 2789	. 2 |- (A.x e. A (ps <-> ch) <-> {x e. A | ps} = {x e. A | ch})
4	2, 3	sylib 188	1 |- (ph -> {x e. A | ps} = {x e. A | ch})

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   = wceq 1642   e. wcel 1710  A.wral 2614  {crab 2618

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-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-ral 2619  df-rab 2623

This theorem is referenced by:  rabbidv  2851  rabeqbidva  2855  rabbi2dva  3463