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Theorem cbvopab1s 3873
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 1462 . . . 4 𝑧𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
2 nfv 1462 . . . . . 6 𝑥 𝑤 = ⟨𝑧, 𝑦
3 nfs1v 1858 . . . . . 6 𝑥[𝑧 / 𝑥]𝜑
42, 3nfan 1498 . . . . 5 𝑥(𝑤 = ⟨𝑧, 𝑦⟩ ∧ [𝑧 / 𝑥]𝜑)
54nfex 1569 . . . 4 𝑥𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ [𝑧 / 𝑥]𝜑)
6 opeq1 3590 . . . . . . 7 (𝑥 = 𝑧 → ⟨𝑥, 𝑦⟩ = ⟨𝑧, 𝑦⟩)
76eqeq2d 2094 . . . . . 6 (𝑥 = 𝑧 → (𝑤 = ⟨𝑥, 𝑦⟩ ↔ 𝑤 = ⟨𝑧, 𝑦⟩))
8 sbequ12 1696 . . . . . 6 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
97, 8anbi12d 457 . . . . 5 (𝑥 = 𝑧 → ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑤 = ⟨𝑧, 𝑦⟩ ∧ [𝑧 / 𝑥]𝜑)))
109exbidv 1748 . . . 4 (𝑥 = 𝑧 → (∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ [𝑧 / 𝑥]𝜑)))
111, 5, 10cbvex 1681 . . 3 (∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑧𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ [𝑧 / 𝑥]𝜑))
1211abbii 2198 . 2 {𝑤 ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {𝑤 ∣ ∃𝑧𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ [𝑧 / 𝑥]𝜑)}
13 df-opab 3860 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
14 df-opab 3860 . 2 {⟨𝑧, 𝑦⟩ ∣ [𝑧 / 𝑥]𝜑} = {𝑤 ∣ ∃𝑧𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ [𝑧 / 𝑥]𝜑)}
1512, 13, 143eqtr4i 2113 1 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑦⟩ ∣ [𝑧 / 𝑥]𝜑}
Colors of variables: wff set class
Syntax hints:  wa 102   = wceq 1285  wex 1422  [wsb 1687  {cab 2069  cop 3419  {copab 3858
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-un 2986  df-sn 3422  df-pr 3423  df-op 3425  df-opab 3860
This theorem is referenced by: (None)
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