Theorem cbvopab2 4079

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 1528	. . . . . 6 ⊢ Ⅎ𝑧 𝑤 = ⟨𝑥, 𝑦⟩
2		cbvopab2.1	. . . . . 6 ⊢ Ⅎ𝑧𝜑
3	1, 2	nfan 1565	. . . . 5 ⊢ Ⅎ𝑧(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
4		nfv 1528	. . . . . 6 ⊢ Ⅎ𝑦 𝑤 = ⟨𝑥, 𝑧⟩
5		cbvopab2.2	. . . . . 6 ⊢ Ⅎ𝑦𝜓
6	4, 5	nfan 1565	. . . . 5 ⊢ Ⅎ𝑦(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝜓)
7		opeq2 3781	. . . . . . 7 ⊢ (𝑦 = 𝑧 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑧⟩)
8	7	eqeq2d 2189	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝑤 = ⟨𝑥, 𝑦⟩ ↔ 𝑤 = ⟨𝑥, 𝑧⟩))
9		cbvopab2.3	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓))
10	8, 9	anbi12d 473	. . . . 5 ⊢ (𝑦 = 𝑧 → ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝜓)))
11	3, 6, 10	cbvex 1756	. . . 4 ⊢ (∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝜓))
12	11	exbii 1605	. . 3 ⊢ (∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝜓))
13	12	abbii 2293	. 2 ⊢ {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {𝑤 ∣ ∃𝑥∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝜓)}
14		df-opab 4067	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
15		df-opab 4067	. 2 ⊢ {⟨𝑥, 𝑧⟩ ∣ 𝜓} = {𝑤 ∣ ∃𝑥∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝜓)}
16	13, 14, 15	3eqtr4i 2208	1 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑧⟩ ∣ 𝜓}

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353  Ⅎwnf 1460  ∃wex 1492  {cab 2163  ⟨cop 3597  {copab 4065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-un 3135  df-sn 3600  df-pr 3601  df-op 3603  df-opab 4067

This theorem is referenced by:  cbvoprab3  5953