Theorem cbvopab1 5216

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.) Add disjoint variable condition to avoid ax-13 2365. See cbvopab1g 5217 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 17-Jan-2024.)

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 group:   𝑥,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 1909	. . . . 5 ⊢ Ⅎ𝑣∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
2		nfv 1909	. . . . . . 7 ⊢ Ⅎ𝑥 𝑤 = ⟨𝑣, 𝑦⟩
3		nfs1v 2145	. . . . . . 7 ⊢ Ⅎ𝑥[𝑣 / 𝑥]𝜑
4	2, 3	nfan 1894	. . . . . 6 ⊢ Ⅎ𝑥(𝑤 = ⟨𝑣, 𝑦⟩ ∧ [𝑣 / 𝑥]𝜑)
5	4	nfex 2311	. . . . 5 ⊢ Ⅎ𝑥∃𝑦(𝑤 = ⟨𝑣, 𝑦⟩ ∧ [𝑣 / 𝑥]𝜑)
6		opeq1 4868	. . . . . . . 8 ⊢ (𝑥 = 𝑣 → ⟨𝑥, 𝑦⟩ = ⟨𝑣, 𝑦⟩)
7	6	eqeq2d 2737	. . . . . . 7 ⊢ (𝑥 = 𝑣 → (𝑤 = ⟨𝑥, 𝑦⟩ ↔ 𝑤 = ⟨𝑣, 𝑦⟩))
8		sbequ12 2235	. . . . . . 7 ⊢ (𝑥 = 𝑣 → (𝜑 ↔ [𝑣 / 𝑥]𝜑))
9	7, 8	anbi12d 630	. . . . . 6 ⊢ (𝑥 = 𝑣 → ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑤 = ⟨𝑣, 𝑦⟩ ∧ [𝑣 / 𝑥]𝜑)))
10	9	exbidv 1916	. . . . 5 ⊢ (𝑥 = 𝑣 → (∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦(𝑤 = ⟨𝑣, 𝑦⟩ ∧ [𝑣 / 𝑥]𝜑)))
11	1, 5, 10	cbvexv1 2332	. . . 4 ⊢ (∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑣∃𝑦(𝑤 = ⟨𝑣, 𝑦⟩ ∧ [𝑣 / 𝑥]𝜑))
12		nfv 1909	. . . . . . 7 ⊢ Ⅎ𝑧 𝑤 = ⟨𝑣, 𝑦⟩
13		cbvopab1.1	. . . . . . . 8 ⊢ Ⅎ𝑧𝜑
14	13	nfsbv 2317	. . . . . . 7 ⊢ Ⅎ𝑧[𝑣 / 𝑥]𝜑
15	12, 14	nfan 1894	. . . . . 6 ⊢ Ⅎ𝑧(𝑤 = ⟨𝑣, 𝑦⟩ ∧ [𝑣 / 𝑥]𝜑)
16	15	nfex 2311	. . . . 5 ⊢ Ⅎ𝑧∃𝑦(𝑤 = ⟨𝑣, 𝑦⟩ ∧ [𝑣 / 𝑥]𝜑)
17		nfv 1909	. . . . 5 ⊢ Ⅎ𝑣∃𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜓)
18		opeq1 4868	. . . . . . . 8 ⊢ (𝑣 = 𝑧 → ⟨𝑣, 𝑦⟩ = ⟨𝑧, 𝑦⟩)
19	18	eqeq2d 2737	. . . . . . 7 ⊢ (𝑣 = 𝑧 → (𝑤 = ⟨𝑣, 𝑦⟩ ↔ 𝑤 = ⟨𝑧, 𝑦⟩))
20		cbvopab1.2	. . . . . . . 8 ⊢ Ⅎ𝑥𝜓
21		cbvopab1.3	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜓))
22	20, 21	sbhypf 3533	. . . . . . 7 ⊢ (𝑣 = 𝑧 → ([𝑣 / 𝑥]𝜑 ↔ 𝜓))
23	19, 22	anbi12d 630	. . . . . 6 ⊢ (𝑣 = 𝑧 → ((𝑤 = ⟨𝑣, 𝑦⟩ ∧ [𝑣 / 𝑥]𝜑) ↔ (𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜓)))
24	23	exbidv 1916	. . . . 5 ⊢ (𝑣 = 𝑧 → (∃𝑦(𝑤 = ⟨𝑣, 𝑦⟩ ∧ [𝑣 / 𝑥]𝜑) ↔ ∃𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜓)))
25	16, 17, 24	cbvexv1 2332	. . . 4 ⊢ (∃𝑣∃𝑦(𝑤 = ⟨𝑣, 𝑦⟩ ∧ [𝑣 / 𝑥]𝜑) ↔ ∃𝑧∃𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜓))
26	11, 25	bitri 275	. . 3 ⊢ (∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑧∃𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜓))
27	26	abbii 2796	. 2 ⊢ {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {𝑤 ∣ ∃𝑧∃𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜓)}
28		df-opab 5204	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
29		df-opab 5204	. 2 ⊢ {⟨𝑧, 𝑦⟩ ∣ 𝜓} = {𝑤 ∣ ∃𝑧∃𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜓)}
30	27, 28, 29	3eqtr4i 2764	1 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑦⟩ ∣ 𝜓}

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533  ∃wex 1773  Ⅎwnf 1777  [wsb 2059  {cab 2703  ⟨cop 4629  {copab 5203

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-opab 5204

This theorem is referenced by:  cbvopab1vOLD  5221  cbvmptf  5250  phpreu  36985  poimirlem26  37027  mbfposadd  37048