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Theorem rabeqbidv 2704
 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 2701 . . 3 (𝐴 = 𝐵 → {𝑥𝐴𝜓} = {𝑥𝐵𝜓})
31, 2syl 14 . 2 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜓})
4 rabeqbidv.2 . . 3 (𝜑 → (𝜓𝜒))
54rabbidv 2698 . 2 (𝜑 → {𝑥𝐵𝜓} = {𝑥𝐵𝜒})
63, 5eqtrd 2187 1 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104   = wceq 1332  {crab 2436 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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-rab 2441 This theorem is referenced by:  elfvmptrab1  5555  mpoxopoveq  6177  supeq123d  6923  phival  12056  dfphi2  12063  cldval  12438  neifval  12479  cnfval  12533  cnpfval  12534  cnprcl2k  12545  hmeofvalg  12642  ispsmet  12662  ismet  12683  isxmet  12684  blfvalps  12724  cncfval  12898
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