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Theorem plcofph 43705
 Description: Given, a,b and a "definition" for c, c is demonstrated. (Contributed by Jarvin Udandy, 8-Sep-2020.)
Hypotheses
Ref Expression
plcofph.1 (𝜒 ↔ ((((𝜑𝜓) ↔ 𝜑) → (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑))) ∧ (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑))))
plcofph.2 𝜑
plcofph.3 𝜓
Assertion
Ref Expression
plcofph 𝜒

Proof of Theorem plcofph
StepHypRef Expression
1 plcofph.2 . . . . 5 𝜑
2 pm3.24 406 . . . . 5 ¬ (𝜑 ∧ ¬ 𝜑)
31, 2pm3.2i 474 . . . 4 (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑))
43a1i 11 . . 3 (((𝜑𝜓) ↔ 𝜑) → (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑)))
54, 3pm3.2i 474 . 2 ((((𝜑𝜓) ↔ 𝜑) → (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑))) ∧ (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑)))
6 plcofph.1 . . . 4 (𝜒 ↔ ((((𝜑𝜓) ↔ 𝜑) → (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑))) ∧ (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑))))
76bicomi 227 . . 3 (((((𝜑𝜓) ↔ 𝜑) → (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑))) ∧ (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑))) ↔ 𝜒)
87biimpi 219 . 2 (((((𝜑𝜓) ↔ 𝜑) → (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑))) ∧ (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑))) → 𝜒)
95, 8ax-mp 5 1 𝜒
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399 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 This theorem is referenced by:  plvcofph  43707  plvcofphax  43708
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