Theorem pm4.83

Description: Theorem *4.83 of WhiteheadRussell p. 122. (Contributed by NM, 3-Jan-2005)

	Ref	Expression
Assertion	pm4.83	⊢ ( ( ( 𝜑 → 𝜓 ) ∧ ( ¬ 𝜑 → 𝜓 ) ) ↔ 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		exmid	⊢ ( 𝜑 ∨ ¬ 𝜑 )
2	1	a1bi	⊢ ( 𝜓 ↔ ( ( 𝜑 ∨ ¬ 𝜑 ) → 𝜓 ) )
3		jaob	⊢ ( ( ( 𝜑 ∨ ¬ 𝜑 ) → 𝜓 ) ↔ ( ( 𝜑 → 𝜓 ) ∧ ( ¬ 𝜑 → 𝜓 ) ) )
4	2 3	bitr2i	⊢ ( ( ( 𝜑 → 𝜓 ) ∧ ( ¬ 𝜑 → 𝜓 ) ) ↔ 𝜓 )