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Theorem rabeqbidva 2759
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 2751 . 2 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
3 rabeqbidva.1 . . 3 (𝜑𝐴 = 𝐵)
4 rabeq 2755 . . 3 (𝐴 = 𝐵 → {𝑥𝐴𝜒} = {𝑥𝐵𝜒})
53, 4syl 14 . 2 (𝜑 → {𝑥𝐴𝜒} = {𝑥𝐵𝜒})
62, 5eqtrd 2229 1 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1364  wcel 2167  {crab 2479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rab 2484
This theorem is referenced by: (None)
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