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Theorem opelopabf 5564
Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 5561 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 5549 . 2 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑)
2 opelopabf.1 . . 3 𝐴 ∈ V
3 nfcv 2908 . . . . 5 𝑥𝐵
4 opelopabf.x . . . . 5 𝑥𝜓
53, 4nfsbcw 3826 . . . 4 𝑥[𝐵 / 𝑦]𝜓
6 opelopabf.3 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
76sbcbidv 3864 . . . 4 (𝑥 = 𝐴 → ([𝐵 / 𝑦]𝜑[𝐵 / 𝑦]𝜓))
85, 7sbciegf 3844 . . 3 (𝐴 ∈ V → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐵 / 𝑦]𝜓))
92, 8ax-mp 5 . 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 3844 . . 3 (𝐵 ∈ V → ([𝐵 / 𝑦]𝜓𝜒))
1410, 13ax-mp 5 . 2 ([𝐵 / 𝑦]𝜓𝜒)
151, 9, 143bitri 297 1 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1537  wnf 1781  wcel 2108  Vcvv 3488  [wsbc 3804  cop 4654  {copab 5228
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-opab 5229
This theorem is referenced by:  pofun  5626  fmptco  7163  fmptcof2  32675
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