Theorem cbvopab1s 5116

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 1922	. . . 4 ⊢ Ⅎ𝑧∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
2		nfv 1922	. . . . . 6 ⊢ Ⅎ𝑥 𝑤 = ⟨𝑧, 𝑦⟩
3		nfs1v 2159	. . . . . 6 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
4	2, 3	nfan 1907	. . . . 5 ⊢ Ⅎ𝑥(𝑤 = ⟨𝑧, 𝑦⟩ ∧ [𝑧 / 𝑥]𝜑)
5	4	nfex 2325	. . . 4 ⊢ Ⅎ𝑥∃𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ [𝑧 / 𝑥]𝜑)
6		opeq1 4770	. . . . . . 7 ⊢ (𝑥 = 𝑧 → ⟨𝑥, 𝑦⟩ = ⟨𝑧, 𝑦⟩)
7	6	eqeq2d 2747	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑤 = ⟨𝑥, 𝑦⟩ ↔ 𝑤 = ⟨𝑧, 𝑦⟩))
8		sbequ12 2251	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
9	7, 8	anbi12d 634	. . . . 5 ⊢ (𝑥 = 𝑧 → ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑤 = ⟨𝑧, 𝑦⟩ ∧ [𝑧 / 𝑥]𝜑)))
10	9	exbidv 1929	. . . 4 ⊢ (𝑥 = 𝑧 → (∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ [𝑧 / 𝑥]𝜑)))
11	1, 5, 10	cbvexv1 2343	. . 3 ⊢ (∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑧∃𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ [𝑧 / 𝑥]𝜑))
12	11	abbii 2801	. 2 ⊢ {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {𝑤 ∣ ∃𝑧∃𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ [𝑧 / 𝑥]𝜑)}
13		df-opab 5102	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
14		df-opab 5102	. 2 ⊢ {⟨𝑧, 𝑦⟩ ∣ [𝑧 / 𝑥]𝜑} = {𝑤 ∣ ∃𝑧∃𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ [𝑧 / 𝑥]𝜑)}
15	12, 13, 14	3eqtr4i 2769	1 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑦⟩ ∣ [𝑧 / 𝑥]𝜑}

Syntax hints:   ∧ wa 399   = wceq 1543  ∃wex 1787  [wsb 2072  {cab 2714  ⟨cop 4533  {copab 5101

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-rab 3060  df-v 3400  df-dif 3856  df-un 3858  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-opab 5102

This theorem is referenced by: (None)