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

Theorem opelopabgf 5545
Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopabg 5543 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 5535 . 2 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑)
2 nfcv 2905 . . . . 5 𝑥𝐵
3 opelopabgf.x . . . . 5 𝑥𝜓
42, 3nfsbcw 3810 . . . 4 𝑥[𝐵 / 𝑦]𝜓
5 opelopabgf.1 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
65sbcbidv 3845 . . . 4 (𝑥 = 𝐴 → ([𝐵 / 𝑦]𝜑[𝐵 / 𝑦]𝜓))
74, 6sbciegf 3827 . . 3 (𝐴𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐵 / 𝑦]𝜓))
8 opelopabgf.y . . . 4 𝑦𝜒
9 opelopabgf.2 . . . 4 (𝑦 = 𝐵 → (𝜓𝜒))
108, 9sbciegf 3827 . . 3 (𝐵𝑊 → ([𝐵 / 𝑦]𝜓𝜒))
117, 10sylan9bb 509 . 2 ((𝐴𝑉𝐵𝑊) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑𝜒))
121, 11bitrid 283 1 ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wnf 1783  wcel 2108  [wsbc 3788  cop 4632  {copab 5205
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-opab 5206
This theorem is referenced by:  oprabv  7493
  Copyright terms: Public domain W3C validator