Theorem cbvopab1 3830

Description: Change first bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 6-Oct-2004.) (Revised by Mario Carneiro, 14-Oct-2016.)

Hypotheses

Ref	Expression
cbvopab1.1	⊢ Ⅎ𝑧𝜑
cbvopab1.2	⊢ Ⅎ𝑥𝜓
cbvopab1.3	⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvopab1	⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑦⟩ ∣ 𝜓}

Distinct variable groups:   𝑥,𝑦   𝑦,𝑧

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

Proof of Theorem cbvopab1

Dummy variables 𝑤 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1421	. . . . 5 ⊢ Ⅎ𝑣∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
2		nfv 1421	. . . . . . 7 ⊢ Ⅎ𝑥 𝑤 = ⟨𝑣, 𝑦⟩
3		nfs1v 1815	. . . . . . 7 ⊢ Ⅎ𝑥[𝑣 / 𝑥]𝜑
4	2, 3	nfan 1457	. . . . . 6 ⊢ Ⅎ𝑥(𝑤 = ⟨𝑣, 𝑦⟩ ∧ [𝑣 / 𝑥]𝜑)
5	4	nfex 1528	. . . . 5 ⊢ Ⅎ𝑥∃𝑦(𝑤 = ⟨𝑣, 𝑦⟩ ∧ [𝑣 / 𝑥]𝜑)
6		opeq1 3549	. . . . . . . 8 ⊢ (𝑥 = 𝑣 → ⟨𝑥, 𝑦⟩ = ⟨𝑣, 𝑦⟩)
7	6	eqeq2d 2051	. . . . . . 7 ⊢ (𝑥 = 𝑣 → (𝑤 = ⟨𝑥, 𝑦⟩ ↔ 𝑤 = ⟨𝑣, 𝑦⟩))
8		sbequ12 1654	. . . . . . 7 ⊢ (𝑥 = 𝑣 → (𝜑 ↔ [𝑣 / 𝑥]𝜑))
9	7, 8	anbi12d 442	. . . . . 6 ⊢ (𝑥 = 𝑣 → ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑤 = ⟨𝑣, 𝑦⟩ ∧ [𝑣 / 𝑥]𝜑)))
10	9	exbidv 1706	. . . . 5 ⊢ (𝑥 = 𝑣 → (∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦(𝑤 = ⟨𝑣, 𝑦⟩ ∧ [𝑣 / 𝑥]𝜑)))
11	1, 5, 10	cbvex 1639	. . . 4 ⊢ (∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑣∃𝑦(𝑤 = ⟨𝑣, 𝑦⟩ ∧ [𝑣 / 𝑥]𝜑))
12		nfv 1421	. . . . . . 7 ⊢ Ⅎ𝑧 𝑤 = ⟨𝑣, 𝑦⟩
13		cbvopab1.1	. . . . . . . 8 ⊢ Ⅎ𝑧𝜑
14	13	nfsb 1822	. . . . . . 7 ⊢ Ⅎ𝑧[𝑣 / 𝑥]𝜑
15	12, 14	nfan 1457	. . . . . 6 ⊢ Ⅎ𝑧(𝑤 = ⟨𝑣, 𝑦⟩ ∧ [𝑣 / 𝑥]𝜑)
16	15	nfex 1528	. . . . 5 ⊢ Ⅎ𝑧∃𝑦(𝑤 = ⟨𝑣, 𝑦⟩ ∧ [𝑣 / 𝑥]𝜑)
17		nfv 1421	. . . . 5 ⊢ Ⅎ𝑣∃𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜓)
18		opeq1 3549	. . . . . . . 8 ⊢ (𝑣 = 𝑧 → ⟨𝑣, 𝑦⟩ = ⟨𝑧, 𝑦⟩)
19	18	eqeq2d 2051	. . . . . . 7 ⊢ (𝑣 = 𝑧 → (𝑤 = ⟨𝑣, 𝑦⟩ ↔ 𝑤 = ⟨𝑧, 𝑦⟩))
20		sbequ 1721	. . . . . . . 8 ⊢ (𝑣 = 𝑧 → ([𝑣 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑))
21		cbvopab1.2	. . . . . . . . 9 ⊢ Ⅎ𝑥𝜓
22		cbvopab1.3	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜓))
23	21, 22	sbie 1674	. . . . . . . 8 ⊢ ([𝑧 / 𝑥]𝜑 ↔ 𝜓)
24	20, 23	syl6bb 185	. . . . . . 7 ⊢ (𝑣 = 𝑧 → ([𝑣 / 𝑥]𝜑 ↔ 𝜓))
25	19, 24	anbi12d 442	. . . . . 6 ⊢ (𝑣 = 𝑧 → ((𝑤 = ⟨𝑣, 𝑦⟩ ∧ [𝑣 / 𝑥]𝜑) ↔ (𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜓)))
26	25	exbidv 1706	. . . . 5 ⊢ (𝑣 = 𝑧 → (∃𝑦(𝑤 = ⟨𝑣, 𝑦⟩ ∧ [𝑣 / 𝑥]𝜑) ↔ ∃𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜓)))
27	16, 17, 26	cbvex 1639	. . . 4 ⊢ (∃𝑣∃𝑦(𝑤 = ⟨𝑣, 𝑦⟩ ∧ [𝑣 / 𝑥]𝜑) ↔ ∃𝑧∃𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜓))
28	11, 27	bitri 173	. . 3 ⊢ (∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑧∃𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜓))
29	28	abbii 2153	. 2 ⊢ {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {𝑤 ∣ ∃𝑧∃𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜓)}
30		df-opab 3819	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
31		df-opab 3819	. 2 ⊢ {⟨𝑧, 𝑦⟩ ∣ 𝜓} = {𝑤 ∣ ∃𝑧∃𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜓)}
32	29, 30, 31	3eqtr4i 2070	1 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑦⟩ ∣ 𝜓}

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243  Ⅎwnf 1349  ∃wex 1381  [wsb 1645  {cab 2026  ⟨cop 3378  {copab 3817

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-opab 3819

This theorem is referenced by:  cbvopab1v  3833  cbvmpt  3851