Theorem rabeqdv 2683

Description: Equality of restricted class abstractions. Deduction form of rabeq 2681. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

Hypothesis

Ref	Expression
rabeqdv.1	|- ( ph -> A = B )

Assertion

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

Distinct variable groups:   x, A   x, B   ph, x

Allowed substitution hint:   ps( x)

Proof of Theorem rabeqdv

Step	Hyp	Ref	Expression
1		rabeqdv.1	. 2 |- ( ph -> A = B )
2		rabeq 2681	. 2 |- ( A = B -> { x e. A | ps } = { x e. B | ps } )
3	1, 2	syl 14	1 |- ( ph -> { x e. A | ps } = { x e. B | ps } )

Syntax hints:   -> wi 4   = wceq 1332   {crab 2421

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-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rab 2426

This theorem is referenced by:  dfphi2  11932  cncfval  12767  reldvg  12856  dvfvalap  12858