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Theorem rabeqbidva 3186
Description: Equality of restricted class abstractions. (Contributed by Mario Carneiro, 26-Jan-2017.)
Hypotheses
Ref Expression
rabeqbidva.1 (𝜑𝐴 = 𝐵)
rabeqbidva.2 ((𝜑𝑥𝐴) → (𝜓𝜒))
Assertion
Ref Expression
rabeqbidva (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)

Proof of Theorem rabeqbidva
StepHypRef Expression
1 rabeqbidva.2 . . 3 ((𝜑𝑥𝐴) → (𝜓𝜒))
21rabbidva 3180 . 2 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
3 rabeqbidva.1 . . 3 (𝜑𝐴 = 𝐵)
43rabeqdv 3184 . 2 (𝜑 → {𝑥𝐴𝜒} = {𝑥𝐵𝜒})
52, 4eqtrd 2655 1 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  {crab 2912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rab 2917
This theorem is referenced by:  natpropd  16576  gsumpropd2lem  17213  elntg  25798  poimirlem28  33108  domnmsuppn0  41468
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