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Theorem opelopabf 4164
Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4161 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 4150 . 2 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑)
2 opelopabf.1 . . 3 𝐴 ∈ V
3 nfcv 2256 . . . . 5 𝑥𝐵
4 opelopabf.x . . . . 5 𝑥𝜓
53, 4nfsbc 2900 . . . 4 𝑥[𝐵 / 𝑦]𝜓
6 opelopabf.3 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
76sbcbidv 2937 . . . 4 (𝑥 = 𝐴 → ([𝐵 / 𝑦]𝜑[𝐵 / 𝑦]𝜓))
85, 7sbciegf 2910 . . 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 2910 . . 3 (𝐵 ∈ V → ([𝐵 / 𝑦]𝜓𝜒))
1410, 13ax-mp 5 . 2 ([𝐵 / 𝑦]𝜓𝜒)
151, 9, 143bitri 205 1 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1314  wnf 1419  wcel 1463  Vcvv 2658  [wsbc 2880  cop 3498  {copab 3956
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-rex 2397  df-v 2660  df-sbc 2881  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-opab 3958
This theorem is referenced by:  pofun  4202  fmptco  5552
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