Theorem rabeqdv 2766

Description: Equality of restricted class abstractions. Deduction form of rabeq 2764. (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

Allowed substitution hints:   ph( x)   ps( x)

Proof of Theorem rabeqdv

Step	Hyp	Ref	Expression
1		rabeqdv.1	. 2 |- ( ph -> A = B )
2		rabeq 2764	. 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 1373   {crab 2488

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-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rab 2493

This theorem is referenced by:  isacnm  7317  elovmpowrd  11037  dfphi2  12575  lspfval  14183  lsppropd  14227  psrval  14461  cncfval  15077  reldvg  15184  dvfvalap  15186