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Theorem rabbidvaOLD 3389
Description: Obsolete proof of rabbidva 3388 as of 4-Dec-2023. (Contributed by NM, 28-Nov-2003.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
rabbidva.1 ((𝜑𝑥𝐴) → (𝜓𝜒))
Assertion
Ref Expression
rabbidvaOLD (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝐴(𝑥)

Proof of Theorem rabbidvaOLD
StepHypRef Expression
1 rabbidva.1 . . 3 ((𝜑𝑥𝐴) → (𝜓𝜒))
21ralrimiva 3105 . 2 (𝜑 → ∀𝑥𝐴 (𝜓𝜒))
3 rabbi 3295 . 2 (∀𝑥𝐴 (𝜓𝜒) ↔ {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
42, 3sylib 221 1 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wcel 2110  wral 3061  {crab 3065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-ral 3066  df-rab 3070
This theorem is referenced by: (None)
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