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Theorem opelopabaf 4153
 Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4151 uses bound-variable hypotheses in place of distinct variable conditions." (Contributed by Mario Carneiro, 19-Dec-2013.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
Hypotheses
Ref Expression
opelopabaf.x 𝑥𝜓
opelopabaf.y 𝑦𝜓
opelopabaf.1 𝐴 ∈ V
opelopabaf.2 𝐵 ∈ V
opelopabaf.3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
Assertion
Ref Expression
opelopabaf (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜓)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem opelopabaf
StepHypRef Expression
1 opelopabsb 4140 . 2 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑)
2 opelopabaf.1 . . 3 𝐴 ∈ V
3 opelopabaf.2 . . 3 𝐵 ∈ V
4 opelopabaf.x . . . 4 𝑥𝜓
5 opelopabaf.y . . . 4 𝑦𝜓
6 nfv 1489 . . . 4 𝑥 𝐵 ∈ V
7 opelopabaf.3 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
84, 5, 6, 7sbc2iegf 2945 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑𝜓))
92, 3, 8mp2an 420 . 2 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑𝜓)
101, 9bitri 183 1 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜓)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1312  Ⅎwnf 1417   ∈ wcel 1461  Vcvv 2655  [wsbc 2876  ⟨cop 3494  {copab 3946 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089 This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-rex 2394  df-v 2657  df-sbc 2877  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-opab 3948 This theorem is referenced by: (None)
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