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Theorem bj-rabeqbida 34645
 Description: Version of rabeqbidva 3399 with two disjoint variable conditions removed and the third replaced by a nonfreeness hypothesis. (Contributed by BJ, 27-Apr-2019.)
Hypotheses
Ref Expression
bj-rabeqbida.nf 𝑥𝜑
bj-rabeqbida.1 (𝜑𝐴 = 𝐵)
bj-rabeqbida.2 ((𝜑𝑥𝐴) → (𝜓𝜒))
Assertion
Ref Expression
bj-rabeqbida (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})

Proof of Theorem bj-rabeqbida
StepHypRef Expression
1 bj-rabeqbida.nf . . 3 𝑥𝜑
2 bj-rabeqbida.2 . . 3 ((𝜑𝑥𝐴) → (𝜓𝜒))
31, 2rabbida 3386 . 2 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
4 bj-rabeqbida.1 . . 3 (𝜑𝐴 = 𝐵)
51, 4bj-rabeqd 34643 . 2 (𝜑 → {𝑥𝐴𝜒} = {𝑥𝐵𝜒})
63, 5eqtrd 2793 1 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  Ⅎwnf 1785   ∈ wcel 2111  {crab 3074 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-rab 3079 This theorem is referenced by: (None)
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