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Theorem opelopabgf 5533
Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopabg 5531 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 5523 . 2 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑)
2 nfcv 2897 . . . . 5 𝑥𝐵
3 opelopabgf.x . . . . 5 𝑥𝜓
42, 3nfsbcw 3794 . . . 4 𝑥[𝐵 / 𝑦]𝜓
5 opelopabgf.1 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
65sbcbidv 3831 . . . 4 (𝑥 = 𝐴 → ([𝐵 / 𝑦]𝜑[𝐵 / 𝑦]𝜓))
74, 6sbciegf 3811 . . 3 (𝐴𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐵 / 𝑦]𝜓))
8 opelopabgf.y . . . 4 𝑦𝜒
9 opelopabgf.2 . . . 4 (𝑦 = 𝐵 → (𝜓𝜒))
108, 9sbciegf 3811 . . 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 205  wa 395   = wceq 1533  wnf 1777  wcel 2098  [wsbc 3772  cop 4629  {copab 5203
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-opab 5204
This theorem is referenced by:  oprabv  7465
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