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Theorem plvcofphax 43708
 Description: Given, a,b,d, and "definitions" for c, e, f, g: g is demonstrated. (Contributed by Jarvin Udandy, 8-Sep-2020.)
Hypotheses
Ref Expression
plvcofphax.1 (𝜒 ↔ ((((𝜑𝜓) ↔ 𝜑) → (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑))) ∧ (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑))))
plvcofphax.2 (𝜏 ↔ ((𝜒𝜃) ∧ (𝜑𝜒) ∧ ((𝜑𝜓) → (𝜓𝜃))))
plvcofphax.3 (𝜂 ↔ (𝜒𝜏))
plvcofphax.4 𝜑
plvcofphax.5 𝜓
plvcofphax.6 𝜃
plvcofphax.7 (𝜁 ↔ ¬ (𝜓 ∧ ¬ 𝜏))
Assertion
Ref Expression
plvcofphax 𝜁

Proof of Theorem plvcofphax
StepHypRef Expression
1 plvcofphax.5 . . . . 5 𝜓
2 plvcofphax.2 . . . . . 6 (𝜏 ↔ ((𝜒𝜃) ∧ (𝜑𝜒) ∧ ((𝜑𝜓) → (𝜓𝜃))))
3 plvcofphax.4 . . . . . 6 𝜑
4 plvcofphax.1 . . . . . . 7 (𝜒 ↔ ((((𝜑𝜓) ↔ 𝜑) → (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑))) ∧ (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑))))
54, 3, 1plcofph 43705 . . . . . 6 𝜒
6 plvcofphax.6 . . . . . 6 𝜃
72, 3, 1, 5, 6pldofph 43706 . . . . 5 𝜏
81, 7pm3.2i 474 . . . 4 (𝜓𝜏)
9 pm3.4 809 . . . 4 ((𝜓𝜏) → (𝜓𝜏))
108, 9ax-mp 5 . . 3 (𝜓𝜏)
11 iman 405 . . . 4 ((𝜓𝜏) ↔ ¬ (𝜓 ∧ ¬ 𝜏))
1211biimpi 219 . . 3 ((𝜓𝜏) → ¬ (𝜓 ∧ ¬ 𝜏))
1310, 12ax-mp 5 . 2 ¬ (𝜓 ∧ ¬ 𝜏)
14 plvcofphax.7 . . . 4 (𝜁 ↔ ¬ (𝜓 ∧ ¬ 𝜏))
1514bicomi 227 . . 3 (¬ (𝜓 ∧ ¬ 𝜏) ↔ 𝜁)
1615biimpi 219 . 2 (¬ (𝜓 ∧ ¬ 𝜏) → 𝜁)
1713, 16ax-mp 5 1 𝜁
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8 This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1086 This theorem is referenced by: (None)
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