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Theorem plvcofph 41434
Description: Given, a,b,d, and "definitions" for c, e, f: f is demonstrated. (Contributed by Jarvin Udandy, 8-Sep-2020.)
Hypotheses
Ref Expression
plvcofph.1 (𝜒 ↔ ((((𝜑𝜓) ↔ 𝜑) → (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑))) ∧ (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑))))
plvcofph.2 (𝜏 ↔ ((𝜒𝜃) ∧ (𝜑𝜒) ∧ ((𝜑𝜓) → (𝜓𝜃))))
plvcofph.3 (𝜂 ↔ (𝜒𝜏))
plvcofph.4 𝜑
plvcofph.5 𝜓
plvcofph.6 𝜃
Assertion
Ref Expression
plvcofph 𝜂

Proof of Theorem plvcofph
StepHypRef Expression
1 plvcofph.1 . . . 4 (𝜒 ↔ ((((𝜑𝜓) ↔ 𝜑) → (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑))) ∧ (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑))))
2 plvcofph.4 . . . 4 𝜑
3 plvcofph.5 . . . 4 𝜓
41, 2, 3plcofph 41432 . . 3 𝜒
5 plvcofph.2 . . . 4 (𝜏 ↔ ((𝜒𝜃) ∧ (𝜑𝜒) ∧ ((𝜑𝜓) → (𝜓𝜃))))
6 plvcofph.6 . . . 4 𝜃
75, 2, 3, 4, 6pldofph 41433 . . 3 𝜏
84, 7pm3.2i 470 . 2 (𝜒𝜏)
9 plvcofph.3 . . . 4 (𝜂 ↔ (𝜒𝜏))
109bicomi 214 . . 3 ((𝜒𝜏) ↔ 𝜂)
1110biimpi 206 . 2 ((𝜒𝜏) → 𝜂)
128, 11ax-mp 5 1 𝜂
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1054
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 197  df-an 385  df-3an 1056
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator