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Theorem opelopabgf 5392
Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopabg 5390 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 5382 . 2 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑)
2 nfcv 2955 . . . . 5 𝑥𝐵
3 opelopabgf.x . . . . 5 𝑥𝜓
42, 3nfsbcw 3742 . . . 4 𝑥[𝐵 / 𝑦]𝜓
5 opelopabgf.1 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
65sbcbidv 3774 . . . 4 (𝑥 = 𝐴 → ([𝐵 / 𝑦]𝜑[𝐵 / 𝑦]𝜓))
74, 6sbciegf 3757 . . 3 (𝐴𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐵 / 𝑦]𝜓))
8 opelopabgf.y . . . 4 𝑦𝜒
9 opelopabgf.2 . . . 4 (𝑦 = 𝐵 → (𝜓𝜒))
108, 9sbciegf 3757 . . 3 (𝐵𝑊 → ([𝐵 / 𝑦]𝜓𝜒))
117, 10sylan9bb 513 . 2 ((𝐴𝑉𝐵𝑊) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑𝜒))
121, 11syl5bb 286 1 ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wnf 1785  wcel 2111  [wsbc 3720  cop 4531  {copab 5092
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-opab 5093
This theorem is referenced by:  oprabv  7193
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