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Theorem rabeqbidvaOLD 3450
Description: Obsolete version of rabeqbidva 3449 as of 1-Sep-2025. (Contributed by Mario Carneiro, 26-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
rabeqbidvaOLD.1 (𝜑𝐴 = 𝐵)
rabeqbidvaOLD.2 ((𝜑𝑥𝐴) → (𝜓𝜒))
Assertion
Ref Expression
rabeqbidvaOLD (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)

Proof of Theorem rabeqbidvaOLD
StepHypRef Expression
1 rabeqbidvaOLD.2 . . 3 ((𝜑𝑥𝐴) → (𝜓𝜒))
21rabbidva 3439 . 2 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
3 rabeqbidvaOLD.1 . . 3 (𝜑𝐴 = 𝐵)
43rabeqdv 3448 . 2 (𝜑 → {𝑥𝐴𝜒} = {𝑥𝐵𝜒})
52, 4eqtrd 2774 1 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  {crab 3432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-rab 3433
This theorem is referenced by: (None)
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