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Theorem rabbid 3474
Description: Version of rabbidv 3479 with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by BJ, 27-Apr-2019.)
Hypotheses
Ref Expression
rabbid.n 𝑥𝜑
rabbid.1 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
rabbid (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})

Proof of Theorem rabbid
StepHypRef Expression
1 rabbid.n . 2 𝑥𝜑
2 rabbid.1 . . 3 (𝜑 → (𝜓𝜒))
32adantr 483 . 2 ((𝜑𝑥𝐴) → (𝜓𝜒))
41, 3rabbida 3473 1 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208   = wceq 1530  wnf 1777  wcel 2107  {crab 3140
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-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-ral 3141  df-rab 3145
This theorem is referenced by:  satfv1  32598  bj-rabeqbid  34227  bj-seex  34229
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