This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem pm4.14

Description: Theorem *4.14 of WhiteheadRussell p. 117. Related to con34b . (Contributed by NM, 3-Jan-2005) (Proof shortened by Wolf Lammen, 23-Oct-2012)

		Ref	Expression
	Assertion	pm4.14	⊢ ( ( ( 𝜑 ∧ 𝜓 ) → 𝜒 ) ↔ ( ( 𝜑 ∧ ¬ 𝜒 ) → ¬ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		con34b	⊢ ( ( 𝜓 → 𝜒 ) ↔ ( ¬ 𝜒 → ¬ 𝜓 ) )
2	1	imbi2i	⊢ ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) ↔ ( 𝜑 → ( ¬ 𝜒 → ¬ 𝜓 ) ) )
3		impexp	⊢ ( ( ( 𝜑 ∧ 𝜓 ) → 𝜒 ) ↔ ( 𝜑 → ( 𝜓 → 𝜒 ) ) )
4		impexp	⊢ ( ( ( 𝜑 ∧ ¬ 𝜒 ) → ¬ 𝜓 ) ↔ ( 𝜑 → ( ¬ 𝜒 → ¬ 𝜓 ) ) )
5	2 3 4	3bitr4i	⊢ ( ( ( 𝜑 ∧ 𝜓 ) → 𝜒 ) ↔ ( ( 𝜑 ∧ ¬ 𝜒 ) → ¬ 𝜓 ) )