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Theorem opelopabf 4039
Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4036 uses bound-variable hypotheses in place of distinct variable conditions." (Contributed by NM, 19-Dec-2008.)
Hypotheses
Ref Expression
opelopabf.x 𝑥𝜓
opelopabf.y 𝑦𝜒
opelopabf.1 𝐴 ∈ V
opelopabf.2 𝐵 ∈ V
opelopabf.3 (𝑥 = 𝐴 → (𝜑𝜓))
opelopabf.4 (𝑦 = 𝐵 → (𝜓𝜒))
Assertion
Ref Expression
opelopabf (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)

Proof of Theorem opelopabf
StepHypRef Expression
1 opelopabsb 4025 . 2 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑)
2 opelopabf.1 . . 3 𝐴 ∈ V
3 nfcv 2194 . . . . 5 𝑥𝐵
4 opelopabf.x . . . . 5 𝑥𝜓
53, 4nfsbc 2807 . . . 4 𝑥[𝐵 / 𝑦]𝜓
6 opelopabf.3 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
76sbcbidv 2844 . . . 4 (𝑥 = 𝐴 → ([𝐵 / 𝑦]𝜑[𝐵 / 𝑦]𝜓))
85, 7sbciegf 2817 . . 3 (𝐴 ∈ V → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐵 / 𝑦]𝜓))
92, 8ax-mp 7 . 2 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐵 / 𝑦]𝜓)
10 opelopabf.2 . . 3 𝐵 ∈ V
11 opelopabf.y . . . 4 𝑦𝜒
12 opelopabf.4 . . . 4 (𝑦 = 𝐵 → (𝜓𝜒))
1311, 12sbciegf 2817 . . 3 (𝐵 ∈ V → ([𝐵 / 𝑦]𝜓𝜒))
1410, 13ax-mp 7 . 2 ([𝐵 / 𝑦]𝜓𝜒)
151, 9, 143bitri 199 1 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102   = wceq 1259  wnf 1365  wcel 1409  Vcvv 2574  [wsbc 2787  cop 3406  {copab 3845
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-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-v 2576  df-sbc 2788  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-opab 3847
This theorem is referenced by:  pofun  4077  fmptco  5358
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