Theorem cbvoprab3v 7483

Description: Rule used to change the third bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 8-Oct-2004.) (Revised by David Abernethy, 19-Jun-2012.)

Hypothesis

Ref	Expression
cbvoprab3v.1	⊢ (𝑧 = 𝑤 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvoprab3v	⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ 𝜓}

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

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

Proof of Theorem cbvoprab3v

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeq2 4840	. . . . . . 7 ⊢ (𝑧 = 𝑤 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑤⟩)
2	1	eqeq2d 2741	. . . . . 6 ⊢ (𝑧 = 𝑤 → (𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ 𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑤⟩))
3		cbvoprab3v.1	. . . . . 6 ⊢ (𝑧 = 𝑤 → (𝜑 ↔ 𝜓))
4	2, 3	anbi12d 632	. . . . 5 ⊢ (𝑧 = 𝑤 → ((𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ (𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∧ 𝜓)))
5	4	cbvexvw 2037	. . . 4 ⊢ (∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑤(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∧ 𝜓))
6	5	2exbii 1849	. . 3 ⊢ (∃𝑥∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦∃𝑤(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∧ 𝜓))
7	6	abbii 2797	. 2 ⊢ {𝑣 ∣ ∃𝑥∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)} = {𝑣 ∣ ∃𝑥∃𝑦∃𝑤(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∧ 𝜓)}
8		df-oprab 7393	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑣 ∣ ∃𝑥∃𝑦∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
9		df-oprab 7393	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ 𝜓} = {𝑣 ∣ ∃𝑥∃𝑦∃𝑤(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∧ 𝜓)}
10	7, 8, 9	3eqtr4i 2763	1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ 𝜓}

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779  {cab 2708  ⟨cop 4597  {coprab 7390

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-oprab 7393

This theorem is referenced by: (None)