Theorem exmid2 36257

Description: An excluded middle law. (Contributed by Giovanni Mascellani, 23-May-2019.)

Hypotheses

Ref	Expression
exmid2.1	⊢ ((𝜓 ∧ 𝜑) → 𝜒)
exmid2.2	⊢ ((¬ 𝜓 ∧ 𝜂) → 𝜒)

Assertion

Ref	Expression
exmid2	⊢ ((𝜑 ∧ 𝜂) → 𝜒)

Proof of Theorem exmid2

Step	Hyp	Ref	Expression
1		simpl 483	. . . . 5 ⊢ ((𝜑 ∧ 𝜂) → 𝜑)
2	1	anim2i 617	. . . 4 ⊢ ((𝜓 ∧ (𝜑 ∧ 𝜂)) → (𝜓 ∧ 𝜑))
3	2	ancoms 459	. . 3 ⊢ (((𝜑 ∧ 𝜂) ∧ 𝜓) → (𝜓 ∧ 𝜑))
4		exmid2.1	. . 3 ⊢ ((𝜓 ∧ 𝜑) → 𝜒)
5	3, 4	syl 17	. 2 ⊢ (((𝜑 ∧ 𝜂) ∧ 𝜓) → 𝜒)
6		simpr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝜂) → 𝜂)
7	6	anim2i 617	. . . 4 ⊢ ((¬ 𝜓 ∧ (𝜑 ∧ 𝜂)) → (¬ 𝜓 ∧ 𝜂))
8	7	ancoms 459	. . 3 ⊢ (((𝜑 ∧ 𝜂) ∧ ¬ 𝜓) → (¬ 𝜓 ∧ 𝜂))
9		exmid2.2	. . 3 ⊢ ((¬ 𝜓 ∧ 𝜂) → 𝜒)
10	8, 9	syl 17	. 2 ⊢ (((𝜑 ∧ 𝜂) ∧ ¬ 𝜓) → 𝜒)
11	5, 10	pm2.61dan 810	1 ⊢ ((𝜑 ∧ 𝜂) → 𝜒)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396

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 206  df-an 397

This theorem is referenced by: (None)