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Theorem rabeqbidva 2682
 Description: Equality of restricted class abstractions. (Contributed by Mario Carneiro, 26-Jan-2017.)
Hypotheses
Ref Expression
rabeqbidva.1 (𝜑𝐴 = 𝐵)
rabeqbidva.2 ((𝜑𝑥𝐴) → (𝜓𝜒))
Assertion
Ref Expression
rabeqbidva (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)

Proof of Theorem rabeqbidva
StepHypRef Expression
1 rabeqbidva.2 . . 3 ((𝜑𝑥𝐴) → (𝜓𝜒))
21rabbidva 2674 . 2 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
3 rabeqbidva.1 . . 3 (𝜑𝐴 = 𝐵)
4 rabeq 2678 . . 3 (𝐴 = 𝐵 → {𝑥𝐴𝜒} = {𝑥𝐵𝜒})
53, 4syl 14 . 2 (𝜑 → {𝑥𝐴𝜒} = {𝑥𝐵𝜒})
62, 5eqtrd 2172 1 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1331   ∈ wcel 1480  {crab 2420 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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rab 2425 This theorem is referenced by: (None)
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