Theorem rabbidva2 2805

Description: Equivalent wff's yield equal restricted class abstractions. (Contributed by Thierry Arnoux, 4-Feb-2017.)

Hypothesis

Ref	Expression
rabbidva2.1	|- ( ph -> ( ( x e. A /\ ps ) <-> ( x e. B /\ ch ) ) )

Assertion

Ref	Expression
rabbidva2	|- ( ph -> { x e. A | ps } = { x e. B | ch } )

Distinct variable group:   ph, x

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

Proof of Theorem rabbidva2

Step	Hyp	Ref	Expression
1		rabbidva2.1	. . 3 |- ( ph -> ( ( x e. A /\ ps ) <-> ( x e. B /\ ch ) ) )
2	1	abbidv 2358	. 2 |- ( ph -> { x | ( x e. A /\ ps ) } = { x | ( x e. B /\ ch ) } )
3		df-rab 2537	. 2 |- { x e. A | ps } = { x | ( x e. A /\ ps ) }
4		df-rab 2537	. 2 |- { x e. B | ch } = { x | ( x e. B /\ ch ) }
5	2, 3, 4	3eqtr4g 2296	1 |- ( ph -> { x e. A | ps } = { x e. B | ch } )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   {cab 2224   {crab 2532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-rab 2537

This theorem is referenced by:  rabbia2  2806  infssfzcldc  10650  infssfzledc  10651  sseqn  11260  hashfibclem  11263