Theorem cbvopab2 5230

Description: Change second bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 22-Aug-2013.)

Hypotheses

Ref	Expression
cbvopab2.1	⊢ Ⅎ𝑧𝜑
cbvopab2.2	⊢ Ⅎ𝑦𝜓
cbvopab2.3	⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvopab2	⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑧⟩ ∣ 𝜓}

Distinct variable group:   𝑥,𝑦,𝑧

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

Proof of Theorem cbvopab2

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1910	. . . . . 6 ⊢ Ⅎ𝑧 𝑤 = ⟨𝑥, 𝑦⟩
2		cbvopab2.1	. . . . . 6 ⊢ Ⅎ𝑧𝜑
3	1, 2	nfan 1895	. . . . 5 ⊢ Ⅎ𝑧(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
4		nfv 1910	. . . . . 6 ⊢ Ⅎ𝑦 𝑤 = ⟨𝑥, 𝑧⟩
5		cbvopab2.2	. . . . . 6 ⊢ Ⅎ𝑦𝜓
6	4, 5	nfan 1895	. . . . 5 ⊢ Ⅎ𝑦(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝜓)
7		opeq2 4880	. . . . . . 7 ⊢ (𝑦 = 𝑧 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑧⟩)
8	7	eqeq2d 2737	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝑤 = ⟨𝑥, 𝑦⟩ ↔ 𝑤 = ⟨𝑥, 𝑧⟩))
9		cbvopab2.3	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓))
10	8, 9	anbi12d 630	. . . . 5 ⊢ (𝑦 = 𝑧 → ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝜓)))
11	3, 6, 10	cbvexv1 2333	. . . 4 ⊢ (∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝜓))
12	11	exbii 1843	. . 3 ⊢ (∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝜓))
13	12	abbii 2796	. 2 ⊢ {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {𝑤 ∣ ∃𝑥∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝜓)}
14		df-opab 5216	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
15		df-opab 5216	. 2 ⊢ {⟨𝑥, 𝑧⟩ ∣ 𝜓} = {𝑤 ∣ ∃𝑥∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝜓)}
16	13, 14, 15	3eqtr4i 2764	1 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑧⟩ ∣ 𝜓}

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1534  ∃wex 1774  Ⅎwnf 1778  {cab 2703  ⟨cop 4639  {copab 5215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-11 2147  ax-12 2167  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-opab 5216

This theorem is referenced by:  cbvoprab3  7516