Theorem cbvopab1davw 36298

Description: Change the first bound variable in an ordered-pair class abstraction. Deduction form. (Contributed by GG, 14-Aug-2025.)

Hypothesis

Ref	Expression
cbvopab1davw.1	⊢ ((𝜑 ∧ 𝑥 = 𝑧) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
cbvopab1davw	⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {⟨𝑧, 𝑦⟩ ∣ 𝜒})

Distinct variable groups:   𝜑,𝑥,𝑦,𝑧   𝜓,𝑧   𝜒,𝑥

Allowed substitution hints:   𝜓(𝑥,𝑦)   𝜒(𝑦,𝑧)

Proof of Theorem cbvopab1davw

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeq1 4820	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → ⟨𝑥, 𝑦⟩ = ⟨𝑧, 𝑦⟩)
2	1	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑧) → ⟨𝑥, 𝑦⟩ = ⟨𝑧, 𝑦⟩)
3	2	eqeq2d 2742	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑧) → (𝑡 = ⟨𝑥, 𝑦⟩ ↔ 𝑡 = ⟨𝑧, 𝑦⟩))
4		cbvopab1davw.1	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑧) → (𝜓 ↔ 𝜒))
5	3, 4	anbi12d 632	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑧) → ((𝑡 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ (𝑡 = ⟨𝑧, 𝑦⟩ ∧ 𝜒)))
6	5	exbidv 1922	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑧) → (∃𝑦(𝑡 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ ∃𝑦(𝑡 = ⟨𝑧, 𝑦⟩ ∧ 𝜒)))
7	6	cbvexdvaw 2040	. . 3 ⊢ (𝜑 → (∃𝑥∃𝑦(𝑡 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ ∃𝑧∃𝑦(𝑡 = ⟨𝑧, 𝑦⟩ ∧ 𝜒)))
8	7	abbidv 2797	. 2 ⊢ (𝜑 → {𝑡 ∣ ∃𝑥∃𝑦(𝑡 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)} = {𝑡 ∣ ∃𝑧∃𝑦(𝑡 = ⟨𝑧, 𝑦⟩ ∧ 𝜒)})
9		df-opab 5149	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {𝑡 ∣ ∃𝑥∃𝑦(𝑡 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)}
10		df-opab 5149	. 2 ⊢ {⟨𝑧, 𝑦⟩ ∣ 𝜒} = {𝑡 ∣ ∃𝑧∃𝑦(𝑡 = ⟨𝑧, 𝑦⟩ ∧ 𝜒)}
11	8, 9, 10	3eqtr4g 2791	1 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {⟨𝑧, 𝑦⟩ ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541  ∃wex 1780  {cab 2709  ⟨cop 4577  {copab 5148

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-opab 5149

This theorem is referenced by:  cbvmptdavw  36301  cbvmptdavw2  36322