Theorem cbvoprab12v 7510

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 4841	. . . . . . . 8 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑣) → ⟨𝑥, 𝑦⟩ = ⟨𝑤, 𝑣⟩)
2	1	opeq1d 4845	. . . . . . 7 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑣) → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑤, 𝑣⟩, 𝑧⟩)
3	2	eqeq2d 2771	. . . . . 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 644	. . . . 5 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑣) → ((𝑢 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ (𝑢 = ⟨⟨𝑤, 𝑣⟩, 𝑧⟩ ∧ 𝜓)))
6	5	exbidv 1953	. . . 4 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑣) → (∃𝑧(𝑢 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑧(𝑢 = ⟨⟨𝑤, 𝑣⟩, 𝑧⟩ ∧ 𝜓)))
7	6	cbvex2vw 2073	. . 3 ⊢ (∃𝑥∃𝑦∃𝑧(𝑢 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑤∃𝑣∃𝑧(𝑢 = ⟨⟨𝑤, 𝑣⟩, 𝑧⟩ ∧ 𝜓))
8	7	abbii 2827	. 2 ⊢ {𝑢 ∣ ∃𝑥∃𝑦∃𝑧(𝑢 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)} = {𝑢 ∣ ∃𝑤∃𝑣∃𝑧(𝑢 = ⟨⟨𝑤, 𝑣⟩, 𝑧⟩ ∧ 𝜓)}
9		df-oprab 7424	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑢 ∣ ∃𝑥∃𝑦∃𝑧(𝑢 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
10		df-oprab 7424	. 2 ⊢ {⟨⟨𝑤, 𝑣⟩, 𝑧⟩ ∣ 𝜓} = {𝑢 ∣ ∃𝑤∃𝑣∃𝑧(𝑢 = ⟨⟨𝑤, 𝑣⟩, 𝑧⟩ ∧ 𝜓)}
11	8, 9, 10	3eqtr4i 2793	1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑤, 𝑣⟩, 𝑧⟩ ∣ 𝜓}

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569  ∃wex 1811  {cab 2738  ⟨cop 4834  {coprab 7421

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-oprab 7424

This theorem is used by:  cbvmpov  7515  cpnnen  16402  cbvmpovw2  37123