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Theorem bj-rabbida 32889
Description: Version of rabbidva 3183 with dv condition replaced by non-freeness hypothesis. (Contributed by BJ, 27-Apr-2019.)
Hypotheses
Ref Expression
bj-rabbida.nf 𝑥𝜑
bj-rabbida.1 ((𝜑𝑥𝐴) → (𝜓𝜒))
Assertion
Ref Expression
bj-rabbida (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})

Proof of Theorem bj-rabbida
StepHypRef Expression
1 bj-rabbida.nf . . 3 𝑥𝜑
2 bj-rabbida.1 . . . 4 ((𝜑𝑥𝐴) → (𝜓𝜒))
32ex 450 . . 3 (𝜑 → (𝑥𝐴 → (𝜓𝜒)))
41, 3ralrimi 2954 . 2 (𝜑 → ∀𝑥𝐴 (𝜓𝜒))
5 rabbi 3115 . 2 (∀𝑥𝐴 (𝜓𝜒) ↔ {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
64, 5sylib 208 1 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wnf 1706  wcel 1988  wral 2909  {crab 2913
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-ral 2914  df-rab 2918
This theorem is referenced by:  bj-rabbid  32890  bj-rabeqbida  32893
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