Theorem rabeqdv 2724

Description: Equality of restricted class abstractions. Deduction form of rabeq 2722. (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 2722	. 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 1348   {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-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  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-clel 2166  df-nfc 2301  df-rab 2457

This theorem is referenced by:  dfphi2  12174  cncfval  13353  reldvg  13442  dvfvalap  13444