Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  opelopabgf Structured version   Visualization version   GIF version

Theorem opelopabgf 4920
 Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopabg 4918 uses bound-variable hypotheses in place of distinct variable conditions. (Contributed by Alexander van der Vekens, 8-Jul-2018.)
Hypotheses
Ref Expression
opelopabgf.x 𝑥𝜓
opelopabgf.y 𝑦𝜒
opelopabgf.1 (𝑥 = 𝐴 → (𝜑𝜓))
opelopabgf.2 (𝑦 = 𝐵 → (𝜓𝜒))
Assertion
Ref Expression
opelopabgf ((𝐴𝑉𝐵𝑊) → (⟨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)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem opelopabgf
StepHypRef Expression
1 opelopabsb 4910 . 2 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑)
2 nfcv 2751 . . . . 5 𝑥𝐵
3 opelopabgf.x . . . . 5 𝑥𝜓
42, 3nfsbc 3424 . . . 4 𝑥[𝐵 / 𝑦]𝜓
5 opelopabgf.1 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
65sbcbidv 3457 . . . 4 (𝑥 = 𝐴 → ([𝐵 / 𝑦]𝜑[𝐵 / 𝑦]𝜓))
74, 6sbciegf 3434 . . 3 (𝐴𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐵 / 𝑦]𝜓))
8 opelopabgf.y . . . 4 𝑦𝜒
9 opelopabgf.2 . . . 4 (𝑦 = 𝐵 → (𝜓𝜒))
108, 9sbciegf 3434 . . 3 (𝐵𝑊 → ([𝐵 / 𝑦]𝜓𝜒))
117, 10sylan9bb 732 . 2 ((𝐴𝑉𝐵𝑊) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑𝜒))
121, 11syl5bb 271 1 ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  Ⅎwnf 1699   ∈ wcel 1977  [wsbc 3402  ⟨cop 4131  {copab 4642 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-opab 4644 This theorem is referenced by:  oprabv  6601
 Copyright terms: Public domain W3C validator