Theorem cbvopab1s 5149

Description: Change first bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 31-Jul-2003.)

Assertion

Ref	Expression
cbvopab1s	⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑦⟩ ∣ [𝑧 / 𝑥]𝜑}

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

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem cbvopab1s

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1921	. . . 4 ⊢ Ⅎ𝑧∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
2		nfv 1921	. . . . . 6 ⊢ Ⅎ𝑥 𝑤 = ⟨𝑧, 𝑦⟩
3		nfs1v 2167	. . . . . 6 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
4	2, 3	nfan 1906	. . . . 5 ⊢ Ⅎ𝑥(𝑤 = ⟨𝑧, 𝑦⟩ ∧ [𝑧 / 𝑥]𝜑)
5	4	nfex 2333	. . . 4 ⊢ Ⅎ𝑥∃𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ [𝑧 / 𝑥]𝜑)
6		opeq1 4804	. . . . . . 7 ⊢ (𝑥 = 𝑧 → ⟨𝑥, 𝑦⟩ = ⟨𝑧, 𝑦⟩)
7	6	eqeq2d 2750	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑤 = ⟨𝑥, 𝑦⟩ ↔ 𝑤 = ⟨𝑧, 𝑦⟩))
8		sbequ12 2263	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
9	7, 8	anbi12d 638	. . . . 5 ⊢ (𝑥 = 𝑧 → ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑤 = ⟨𝑧, 𝑦⟩ ∧ [𝑧 / 𝑥]𝜑)))
10	9	exbidv 1928	. . . 4 ⊢ (𝑥 = 𝑧 → (∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ [𝑧 / 𝑥]𝜑)))
11	1, 5, 10	cbvexv1 2350	. . 3 ⊢ (∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑧∃𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ [𝑧 / 𝑥]𝜑))
12	11	abbii 2806	. 2 ⊢ {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {𝑤 ∣ ∃𝑧∃𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ [𝑧 / 𝑥]𝜑)}
13		df-opab 5135	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
14		df-opab 5135	. 2 ⊢ {⟨𝑧, 𝑦⟩ ∣ [𝑧 / 𝑥]𝜑} = {𝑤 ∣ ∃𝑧∃𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ [𝑧 / 𝑥]𝜑)}
15	12, 13, 14	3eqtr4i 2772	1 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑦⟩ ∣ [𝑧 / 𝑥]𝜑}

Syntax hints:   ∧ wa 396   = wceq 1547  ∃wex 1786  [wsb 2073  {cab 2717  ⟨cop 4561  {copab 5134

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-opab 5135

This theorem is referenced by: (None)