Theorem rabeqdv 2794

Description: Equality of restricted class abstractions. Deduction form of rabeq 2792. (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 2792	. 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  7408  elovmpowrd  11145  dfphi2  12782  lspfval  14392  lsppropd  14436  psrval  14670  cncfval  15286  reldvg  15393  dvfvalap  15395  isuhgrm  15912  isushgrm  15913  uhgreq12g  15917  isuhgropm  15922  uhgr0vb  15925  uhgrun  15927  isupgren  15936  upgrop  15945  isumgren  15946  upgrun  15965  umgrun  15967  isuspgren  15996  isusgren  15997  isuspgropen  16003  isusgropen  16004  isausgren  16006  ausgrusgrben  16007  usgrstrrepeen  16070  vtxdgfi0e  16101  1loopgrvd2fi  16111  clwwlknonmpo  16223  clwwlknon  16224  clwwlk0on0  16226