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Theorem rabeqif 3404
 Description: Equality theorem for restricted class abstractions. Inference form of rabeqf 3403. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
rabeqf.1 𝑥𝐴
rabeqf.2 𝑥𝐵
rabeqif.3 𝐴 = 𝐵
Assertion
Ref Expression
rabeqif {𝑥𝐴𝜑} = {𝑥𝐵𝜑}

Proof of Theorem rabeqif
StepHypRef Expression
1 rabeqif.3 . 2 𝐴 = 𝐵
2 rabeqf.1 . . 3 𝑥𝐴
3 rabeqf.2 . . 3 𝑥𝐵
42, 3rabeqf 3403 . 2 (𝐴 = 𝐵 → {𝑥𝐴𝜑} = {𝑥𝐵𝜑})
51, 4ax-mp 5 1 {𝑥𝐴𝜑} = {𝑥𝐵𝜑}
 Colors of variables: wff setvar class Syntax hints:   = wceq 1656  Ⅎwnfc 2956  {crab 3121 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-ext 2803 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-rab 3126 This theorem is referenced by:  rabeqi  3406  rabrabi  3413  extwwlkfab  27736  extwwlkfabOLD  27737  smfliminf  41825
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