Theorem rabbidva2 2713

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 2284	. 2 |- ( ph -> { x | ( x e. A /\ ps ) } = { x | ( x e. B /\ ch ) } )
3		df-rab 2453	. 2 |- { x e. A | ps } = { x | ( x e. A /\ ps ) }
4		df-rab 2453	. 2 |- { x e. B | ch } = { x | ( x e. B /\ ch ) }
5	2, 3, 4	3eqtr4g 2224	1 |- ( ph -> { x e. A | ps } = { x e. B | ch } )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   {cab 2151   {crab 2448

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-rab 2453

This theorem is referenced by: (None)