Theorem rabbidva2 2717

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 2288	. 2 |- ( ph -> { x | ( x e. A /\ ps ) } = { x | ( x e. B /\ ch ) } )
3		df-rab 2457	. 2 |- { x e. A | ps } = { x | ( x e. A /\ ps ) }
4		df-rab 2457	. 2 |- { x e. B | ch } = { x | ( x e. B /\ ch ) }
5	2, 3, 4	3eqtr4g 2228	1 |- ( ph -> { x e. A | ps } = { x e. B | ch } )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   {cab 2156   {crab 2452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-rab 2457

This theorem is referenced by: (None)