Theorem cbvoprab2davw 36672

Description: Change the second bound variable in an operation abstraction. Deduction form. (Contributed by GG, 14-Aug-2025.)

Hypothesis

Ref	Expression
cbvoprab2davw.1	⊢ ((𝜑 ∧ 𝑦 = 𝑤) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
cbvoprab2davw	⊢ (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∣ 𝜒})

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

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

Proof of Theorem cbvoprab2davw

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeq2 4843	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑤⟩)
2	1	adantl 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 = 𝑤) → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑤⟩)
3	2	opeq1d 4848	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 = 𝑤) → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩)
4	3	eqeq2d 2780	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 = 𝑤) → (𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ 𝑡 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩))
5		cbvoprab2davw.1	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 = 𝑤) → (𝜓 ↔ 𝜒))
6	4, 5	anbi12d 643	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝑤) → ((𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ (𝑡 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∧ 𝜒)))
7	6	exbidv 1948	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 𝑤) → (∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑧(𝑡 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∧ 𝜒)))
8	7	cbvexdvaw 2066	. . . 4 ⊢ (𝜑 → (∃𝑦∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑤∃𝑧(𝑡 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∧ 𝜒)))
9	8	exbidv 1948	. . 3 ⊢ (𝜑 → (∃𝑥∃𝑦∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑥∃𝑤∃𝑧(𝑡 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∧ 𝜒)))
10	9	abbidv 2835	. 2 ⊢ (𝜑 → {𝑡 ∣ ∃𝑥∃𝑦∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)} = {𝑡 ∣ ∃𝑥∃𝑤∃𝑧(𝑡 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∧ 𝜒)})
11		df-oprab 7415	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {𝑡 ∣ ∃𝑥∃𝑦∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)}
12		df-oprab 7415	. 2 ⊢ {⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∣ 𝜒} = {𝑡 ∣ ∃𝑥∃𝑤∃𝑧(𝑡 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∧ 𝜒)}
13	10, 11, 12	3eqtr4g 2829	1 ⊢ (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567  ∃wex 1806  {cab 2747  ⟨cop 4600  {coprab 7412

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-oprab 7415

This theorem is referenced by:  cbvmpo2davw2  36693