Theorem rabeqdv 2793

Description: Equality of restricted class abstractions. Deduction form of rabeq 2791. (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 2791	. 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 1395   {crab 2512

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rab 2517

This theorem is referenced by:  isacnm  7385  elovmpowrd  11113  dfphi2  12742  lspfval  14352  lsppropd  14396  psrval  14630  cncfval  15246  reldvg  15353  dvfvalap  15355  isuhgrm  15871  isushgrm  15872  uhgreq12g  15876  isuhgropm  15881  uhgr0vb  15884  uhgrun  15886  isupgren  15895  upgrop  15904  isumgren  15905  upgrun  15924  umgrun  15926  isuspgren  15955  isusgren  15956  isuspgropen  15962  isusgropen  15963  isausgren  15965  ausgrusgrben  15966  usgrstrrepeen  16029