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Theorem fnopabeqd 32684
Description: Equality deduction for function abstractions. (Contributed by Jeff Madsen, 19-Jun-2011.)
Hypothesis
Ref Expression
fnopabeqd.1 (𝜑𝐵 = 𝐶)
Assertion
Ref Expression
fnopabeqd (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)})
Distinct variable groups:   𝜑,𝑥   𝜑,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)

Proof of Theorem fnopabeqd
StepHypRef Expression
1 fnopabeqd.1 . . . 4 (𝜑𝐵 = 𝐶)
21eqeq2d 2620 . . 3 (𝜑 → (𝑦 = 𝐵𝑦 = 𝐶))
32anbi2d 736 . 2 (𝜑 → ((𝑥𝐴𝑦 = 𝐵) ↔ (𝑥𝐴𝑦 = 𝐶)))
43opabbidv 4648 1 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  {copab 4642
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-opab 4644
This theorem is referenced by: (None)
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