ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  cbvopab2 GIF version

Theorem cbvopab2 4053
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 1515 . . . . . 6 𝑧 𝑤 = ⟨𝑥, 𝑦
2 cbvopab2.1 . . . . . 6 𝑧𝜑
31, 2nfan 1552 . . . . 5 𝑧(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
4 nfv 1515 . . . . . 6 𝑦 𝑤 = ⟨𝑥, 𝑧
5 cbvopab2.2 . . . . . 6 𝑦𝜓
64, 5nfan 1552 . . . . 5 𝑦(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝜓)
7 opeq2 3756 . . . . . . 7 (𝑦 = 𝑧 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑧⟩)
87eqeq2d 2176 . . . . . 6 (𝑦 = 𝑧 → (𝑤 = ⟨𝑥, 𝑦⟩ ↔ 𝑤 = ⟨𝑥, 𝑧⟩))
9 cbvopab2.3 . . . . . 6 (𝑦 = 𝑧 → (𝜑𝜓))
108, 9anbi12d 465 . . . . 5 (𝑦 = 𝑧 → ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝜓)))
113, 6, 10cbvex 1743 . . . 4 (∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝜓))
1211exbii 1592 . . 3 (∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝜓))
1312abbii 2280 . 2 {𝑤 ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {𝑤 ∣ ∃𝑥𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝜓)}
14 df-opab 4041 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
15 df-opab 4041 . 2 {⟨𝑥, 𝑧⟩ ∣ 𝜓} = {𝑤 ∣ ∃𝑥𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝜓)}
1613, 14, 153eqtr4i 2195 1 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑧⟩ ∣ 𝜓}
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1342  wnf 1447  wex 1479  {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:  cbvoprab3  5912
  Copyright terms: Public domain W3C validator