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Theorem rabeqbidv 2614
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 2611 . . 3 (𝐴 = 𝐵 → {𝑥𝐴𝜓} = {𝑥𝐵𝜓})
31, 2syl 14 . 2 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜓})
4 rabeqbidv.2 . . 3 (𝜑 → (𝜓𝜒))
54rabbidv 2608 . 2 (𝜑 → {𝑥𝐵𝜓} = {𝑥𝐵𝜒})
63, 5eqtrd 2120 1 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1289  {crab 2363
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rab 2368
This theorem is referenced by:  mpt2xopoveq  5987  supeq123d  6665  phival  11282  dfphi2  11289
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