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Theorem cbvopab2 3858
Description: Change second bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 22-Aug-2013.)
Hypotheses
Ref Expression
cbvopab2.1 𝑧𝜑
cbvopab2.2 𝑦𝜓
cbvopab2.3 (𝑦 = 𝑧 → (𝜑𝜓))
Assertion
Ref Expression
cbvopab2 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑧⟩ ∣ 𝜓}
Distinct variable group:   𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑥,𝑦,𝑧)

Proof of Theorem cbvopab2
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 nfv 1437 . . . . . 6 𝑧 𝑤 = ⟨𝑥, 𝑦
2 cbvopab2.1 . . . . . 6 𝑧𝜑
31, 2nfan 1473 . . . . 5 𝑧(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
4 nfv 1437 . . . . . 6 𝑦 𝑤 = ⟨𝑥, 𝑧
5 cbvopab2.2 . . . . . 6 𝑦𝜓
64, 5nfan 1473 . . . . 5 𝑦(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝜓)
7 opeq2 3577 . . . . . . 7 (𝑦 = 𝑧 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑧⟩)
87eqeq2d 2067 . . . . . 6 (𝑦 = 𝑧 → (𝑤 = ⟨𝑥, 𝑦⟩ ↔ 𝑤 = ⟨𝑥, 𝑧⟩))
9 cbvopab2.3 . . . . . 6 (𝑦 = 𝑧 → (𝜑𝜓))
108, 9anbi12d 450 . . . . 5 (𝑦 = 𝑧 → ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝜓)))
113, 6, 10cbvex 1655 . . . 4 (∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝜓))
1211exbii 1512 . . 3 (∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝜓))
1312abbii 2169 . 2 {𝑤 ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {𝑤 ∣ ∃𝑥𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝜓)}
14 df-opab 3846 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
15 df-opab 3846 . 2 {⟨𝑥, 𝑧⟩ ∣ 𝜓} = {𝑤 ∣ ∃𝑥𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝜓)}
1613, 14, 153eqtr4i 2086 1 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑧⟩ ∣ 𝜓}
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wnf 1365  wex 1397  {cab 2042  cop 3405  {copab 3844
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2949  df-sn 3408  df-pr 3409  df-op 3411  df-opab 3846
This theorem is referenced by:  cbvoprab3  5607
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