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Theorem rabeqbidv 2653
Description: Equality of restricted class abstractions. (Contributed by Jeff Madsen, 1-Dec-2009.)
Hypotheses
Ref Expression
rabeqbidv.1 (𝜑𝐴 = 𝐵)
rabeqbidv.2 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
rabeqbidv (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)

Proof of Theorem rabeqbidv
StepHypRef Expression
1 rabeqbidv.1 . . 3 (𝜑𝐴 = 𝐵)
2 rabeq 2650 . . 3 (𝐴 = 𝐵 → {𝑥𝐴𝜓} = {𝑥𝐵𝜓})
31, 2syl 14 . 2 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜓})
4 rabeqbidv.2 . . 3 (𝜑 → (𝜓𝜒))
54rabbidv 2647 . 2 (𝜑 → {𝑥𝐵𝜓} = {𝑥𝐵𝜒})
63, 5eqtrd 2148 1 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1314  {crab 2395
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rab 2400
This theorem is referenced by:  elfvmptrab1  5481  mpoxopoveq  6103  supeq123d  6844  phival  11795  dfphi2  11802  cldval  12174  neifval  12215  cnfval  12269  cnpfval  12270  cnprcl2k  12281  hmeofvalg  12378  ispsmet  12398  ismet  12419  isxmet  12420  blfvalps  12460  cncfval  12634
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