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Theorem opelopabf 4011
 Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4008 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 3997 . 2 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑)
2 opelopabf.1 . . 3 𝐴 ∈ V
3 nfcv 2178 . . . . 5 𝑥𝐵
4 opelopabf.x . . . . 5 𝑥𝜓
53, 4nfsbc 2784 . . . 4 𝑥[𝐵 / 𝑦]𝜓
6 opelopabf.3 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
76sbcbidv 2817 . . . 4 (𝑥 = 𝐴 → ([𝐵 / 𝑦]𝜑[𝐵 / 𝑦]𝜓))
85, 7sbciegf 2794 . . 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 2794 . . 3 (𝐵 ∈ V → ([𝐵 / 𝑦]𝜓𝜒))
1410, 13ax-mp 7 . 2 ([𝐵 / 𝑦]𝜓𝜒)
151, 9, 143bitri 195 1 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1243  Ⅎwnf 1349   ∈ wcel 1393  Vcvv 2557  [wsbc 2764  ⟨cop 3378  {copab 3817 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-opab 3819 This theorem is referenced by:  pofun  4049  fmptco  5330
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