Theorem cbvopab1s 4158

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 1574	. . . 4 ⊢ Ⅎ𝑧∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
2		nfv 1574	. . . . . 6 ⊢ Ⅎ𝑥 𝑤 = ⟨𝑧, 𝑦⟩
3		nfs1v 1990	. . . . . 6 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
4	2, 3	nfan 1611	. . . . 5 ⊢ Ⅎ𝑥(𝑤 = ⟨𝑧, 𝑦⟩ ∧ [𝑧 / 𝑥]𝜑)
5	4	nfex 1683	. . . 4 ⊢ Ⅎ𝑥∃𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ [𝑧 / 𝑥]𝜑)
6		opeq1 3856	. . . . . . 7 ⊢ (𝑥 = 𝑧 → ⟨𝑥, 𝑦⟩ = ⟨𝑧, 𝑦⟩)
7	6	eqeq2d 2241	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑤 = ⟨𝑥, 𝑦⟩ ↔ 𝑤 = ⟨𝑧, 𝑦⟩))
8		sbequ12 1817	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
9	7, 8	anbi12d 473	. . . . 5 ⊢ (𝑥 = 𝑧 → ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑤 = ⟨𝑧, 𝑦⟩ ∧ [𝑧 / 𝑥]𝜑)))
10	9	exbidv 1871	. . . 4 ⊢ (𝑥 = 𝑧 → (∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ [𝑧 / 𝑥]𝜑)))
11	1, 5, 10	cbvex 1802	. . 3 ⊢ (∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑧∃𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ [𝑧 / 𝑥]𝜑))
12	11	abbii 2345	. 2 ⊢ {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {𝑤 ∣ ∃𝑧∃𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ [𝑧 / 𝑥]𝜑)}
13		df-opab 4145	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
14		df-opab 4145	. 2 ⊢ {⟨𝑧, 𝑦⟩ ∣ [𝑧 / 𝑥]𝜑} = {𝑤 ∣ ∃𝑧∃𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ [𝑧 / 𝑥]𝜑)}
15	12, 13, 14	3eqtr4i 2260	1 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑦⟩ ∣ [𝑧 / 𝑥]𝜑}

Syntax hints:   ∧ wa 104   = wceq 1395  ∃wex 1538  [wsb 1808  {cab 2215  ⟨cop 3669  {copab 4143

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-opab 4145

This theorem is referenced by: (None)