Theorem plcofph 46961

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 ↔ ((((𝜑 ∧ 𝜓) ↔ 𝜑) → (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑))) ∧ (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑))))
plcofph.2	⊢ 𝜑
plcofph.3	⊢ 𝜓

Assertion

Ref	Expression
plcofph	⊢ 𝜒

Proof of Theorem plcofph

Step	Hyp	Ref	Expression
1		plcofph.2	. . . . 5 ⊢ 𝜑
2		pm3.24 402	. . . . 5 ⊢ ¬ (𝜑 ∧ ¬ 𝜑)
3	1, 2	pm3.2i 470	. . . 4 ⊢ (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑))
4	3	a1i 11	. . 3 ⊢ (((𝜑 ∧ 𝜓) ↔ 𝜑) → (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑)))
5	4, 3	pm3.2i 470	. 2 ⊢ ((((𝜑 ∧ 𝜓) ↔ 𝜑) → (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑))) ∧ (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑)))
6		plcofph.1	. . . 4 ⊢ (𝜒 ↔ ((((𝜑 ∧ 𝜓) ↔ 𝜑) → (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑))) ∧ (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑))))
7	6	bicomi 224	. . 3 ⊢ (((((𝜑 ∧ 𝜓) ↔ 𝜑) → (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑))) ∧ (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑))) ↔ 𝜒)
8	7	biimpi 216	. 2 ⊢ (((((𝜑 ∧ 𝜓) ↔ 𝜑) → (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑))) ∧ (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑))) → 𝜒)
9	5, 8	ax-mp 5	1 ⊢ 𝜒

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395

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 207  df-an 396

This theorem is referenced by:  plvcofph  46963  plvcofphax  46964