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Theorem bj-rabeqbid 32109
Description: Version of rabeqbidv 3168 with two dv conditions removed and the third replaced by a non-freeness hypothesis. (Contributed by BJ, 27-Apr-2019.)
Hypotheses
Ref Expression
bj-rabeqbid.nf 𝑥𝜑
bj-rabeqbid.1 (𝜑𝐴 = 𝐵)
bj-rabeqbid.2 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
bj-rabeqbid (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})

Proof of Theorem bj-rabeqbid
StepHypRef Expression
1 bj-rabeqbid.nf . . 3 𝑥𝜑
2 bj-rabeqbid.1 . . 3 (𝜑𝐴 = 𝐵)
31, 2bj-rabeqd 32108 . 2 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜓})
4 bj-rabeqbid.2 . . 3 (𝜑 → (𝜓𝜒))
51, 4bj-rabbid 32107 . 2 (𝜑 → {𝑥𝐵𝜓} = {𝑥𝐵𝜒})
63, 5eqtrd 2644 1 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195   = wceq 1475  wnf 1699  {crab 2900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-ral 2901  df-rab 2905
This theorem is referenced by: (None)
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