Theorem cbvoprab12v 7458

Description: Rule used to change first two bound variables in an operation abstraction, using implicit substitution. (Contributed by NM, 8-Oct-2004.)

Hypothesis

Ref	Expression
cbvoprab12v.1	⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑣) → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvoprab12v	⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑤, 𝑣⟩, 𝑧⟩ ∣ 𝜓}

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

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

Proof of Theorem cbvoprab12v

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeq12 4833	. . . . . . . 8 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑣) → ⟨𝑥, 𝑦⟩ = ⟨𝑤, 𝑣⟩)
2	1	opeq1d 4837	. . . . . . 7 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑣) → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑤, 𝑣⟩, 𝑧⟩)
3	2	eqeq2d 2748	. . . . . 6 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑣) → (𝑢 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ 𝑢 = ⟨⟨𝑤, 𝑣⟩, 𝑧⟩))
4		cbvoprab12v.1	. . . . . 6 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑣) → (𝜑 ↔ 𝜓))
5	3, 4	anbi12d 633	. . . . 5 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑣) → ((𝑢 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ (𝑢 = ⟨⟨𝑤, 𝑣⟩, 𝑧⟩ ∧ 𝜓)))
6	5	exbidv 1923	. . . 4 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑣) → (∃𝑧(𝑢 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑧(𝑢 = ⟨⟨𝑤, 𝑣⟩, 𝑧⟩ ∧ 𝜓)))
7	6	cbvex2vw 2043	. . 3 ⊢ (∃𝑥∃𝑦∃𝑧(𝑢 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑤∃𝑣∃𝑧(𝑢 = ⟨⟨𝑤, 𝑣⟩, 𝑧⟩ ∧ 𝜓))
8	7	abbii 2804	. 2 ⊢ {𝑢 ∣ ∃𝑥∃𝑦∃𝑧(𝑢 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)} = {𝑢 ∣ ∃𝑤∃𝑣∃𝑧(𝑢 = ⟨⟨𝑤, 𝑣⟩, 𝑧⟩ ∧ 𝜓)}
9		df-oprab 7372	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑢 ∣ ∃𝑥∃𝑦∃𝑧(𝑢 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
10		df-oprab 7372	. 2 ⊢ {⟨⟨𝑤, 𝑣⟩, 𝑧⟩ ∣ 𝜓} = {𝑢 ∣ ∃𝑤∃𝑣∃𝑧(𝑢 = ⟨⟨𝑤, 𝑣⟩, 𝑧⟩ ∧ 𝜓)}
11	8, 9, 10	3eqtr4i 2770	1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑤, 𝑣⟩, 𝑧⟩ ∣ 𝜓}

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  ∃wex 1781  {cab 2715  ⟨cop 4588  {coprab 7369

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-oprab 7372

This theorem is referenced by:  cbvmpov  7463  cpnnen  16166  cbvmpovw2  36455