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Theorem fnopabeqd 35103
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 2835 . . 3 (𝜑 → (𝑦 = 𝐵𝑦 = 𝐶))
32anbi2d 631 . 2 (𝜑 → ((𝑥𝐴𝑦 = 𝐵) ↔ (𝑥𝐴𝑦 = 𝐶)))
43opabbidv 5118 1 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  {copab 5114
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-9 2125  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-opab 5115
This theorem is referenced by: (None)
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