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Theorem bj-rabeqbid 34126
Description: Version of rabeqbidv 3490 with two disjoint variable conditions removed and the third replaced by a nonfreeness 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 34125 . 2 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜓})
4 bj-rabeqbid.2 . . 3 (𝜑 → (𝜓𝜒))
51, 4rabbid 3480 . 2 (𝜑 → {𝑥𝐵𝜓} = {𝑥𝐵𝜒})
63, 5eqtrd 2860 1 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1530  wnf 1777  {crab 3146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2804  df-cleq 2818  df-clel 2897  df-ral 3147  df-rab 3151
This theorem is referenced by: (None)
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