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Theorem rabeqbidv 2569
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 2568 . . 3 (𝐴 = 𝐵 → {𝑥𝐴𝜓} = {𝑥𝐵𝜓})
31, 2syl 14 . 2 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜓})
4 rabeqbidv.2 . . 3 (𝜑 → (𝜓𝜒))
54rabbidv 2566 . 2 (𝜑 → {𝑥𝐵𝜓} = {𝑥𝐵𝜒})
63, 5eqtrd 2088 1 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102   = wceq 1259  {crab 2327
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rab 2332
This theorem is referenced by:  mpt2xopoveq  5886  supeq123d  6397
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