Theorem fnopabeqd 33514

Description: Equality deduction for function abstractions. (Contributed by Jeff Madsen, 19-Jun-2011.)

Hypothesis

Ref	Expression
fnopabeqd.1	⊢ (𝜑 → 𝐵 = 𝐶)

Assertion

Ref	Expression
fnopabeqd	⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)})

Distinct variable groups:   𝜑,𝑥   𝜑,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)

Proof of Theorem fnopabeqd

Step	Hyp	Ref	Expression
1		fnopabeqd.1	. . . 4 ⊢ (𝜑 → 𝐵 = 𝐶)
2	1	eqeq2d 2632	. . 3 ⊢ (𝜑 → (𝑦 = 𝐵 ↔ 𝑦 = 𝐶))
3	2	anbi2d 740	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)))
4	3	opabbidv 4716	1 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)})

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {copab 4712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-opab 4713

This theorem is referenced by: (None)