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Theorem opelopabgf 5145
Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopabg 5143 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 5135 . 2 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑)
2 nfcv 2902 . . . . 5 𝑥𝐵
3 opelopabgf.x . . . . 5 𝑥𝜓
42, 3nfsbc 3598 . . . 4 𝑥[𝐵 / 𝑦]𝜓
5 opelopabgf.1 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
65sbcbidv 3631 . . . 4 (𝑥 = 𝐴 → ([𝐵 / 𝑦]𝜑[𝐵 / 𝑦]𝜓))
74, 6sbciegf 3608 . . 3 (𝐴𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐵 / 𝑦]𝜓))
8 opelopabgf.y . . . 4 𝑦𝜒
9 opelopabgf.2 . . . 4 (𝑦 = 𝐵 → (𝜓𝜒))
108, 9sbciegf 3608 . . 3 (𝐵𝑊 → ([𝐵 / 𝑦]𝜓𝜒))
117, 10sylan9bb 738 . 2 ((𝐴𝑉𝐵𝑊) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑𝜒))
121, 11syl5bb 272 1 ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wnf 1857  wcel 2139  [wsbc 3576  cop 4327  {copab 4864
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-opab 4865
This theorem is referenced by:  oprabv  6869
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