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Theorem cbvopab1s 4054
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.
StepHypRef Expression
1 nfv 1515 . . . 4 𝑧𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
2 nfv 1515 . . . . . 6 𝑥 𝑤 = ⟨𝑧, 𝑦
3 nfs1v 1926 . . . . . 6 𝑥[𝑧 / 𝑥]𝜑
42, 3nfan 1552 . . . . 5 𝑥(𝑤 = ⟨𝑧, 𝑦⟩ ∧ [𝑧 / 𝑥]𝜑)
54nfex 1624 . . . 4 𝑥𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ [𝑧 / 𝑥]𝜑)
6 opeq1 3755 . . . . . . 7 (𝑥 = 𝑧 → ⟨𝑥, 𝑦⟩ = ⟨𝑧, 𝑦⟩)
76eqeq2d 2176 . . . . . 6 (𝑥 = 𝑧 → (𝑤 = ⟨𝑥, 𝑦⟩ ↔ 𝑤 = ⟨𝑧, 𝑦⟩))
8 sbequ12 1758 . . . . . 6 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
97, 8anbi12d 465 . . . . 5 (𝑥 = 𝑧 → ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑤 = ⟨𝑧, 𝑦⟩ ∧ [𝑧 / 𝑥]𝜑)))
109exbidv 1812 . . . 4 (𝑥 = 𝑧 → (∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ [𝑧 / 𝑥]𝜑)))
111, 5, 10cbvex 1743 . . 3 (∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑧𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ [𝑧 / 𝑥]𝜑))
1211abbii 2280 . 2 {𝑤 ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {𝑤 ∣ ∃𝑧𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ [𝑧 / 𝑥]𝜑)}
13 df-opab 4041 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
14 df-opab 4041 . 2 {⟨𝑧, 𝑦⟩ ∣ [𝑧 / 𝑥]𝜑} = {𝑤 ∣ ∃𝑧𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ [𝑧 / 𝑥]𝜑)}
1512, 13, 143eqtr4i 2195 1 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑦⟩ ∣ [𝑧 / 𝑥]𝜑}
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1342  wex 1479  [wsb 1749  {cab 2150  cop 3576  {copab 4039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2726  df-un 3118  df-sn 3579  df-pr 3580  df-op 3582  df-opab 4041
This theorem is referenced by: (None)
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