Theorem pm4.87

Description: Theorem *4.87 of WhiteheadRussell p. 122. (Contributed by NM, 3-Jan-2005) (Proof shortened by Eric Schmidt, 26-Oct-2006)

	Ref	Expression
Assertion	pm4.87	⊢ ( ( ( ( ( 𝜑 ∧ 𝜓 ) → 𝜒 ) ↔ ( 𝜑 → ( 𝜓 → 𝜒 ) ) ) ∧ ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) ↔ ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ∧ ( ( 𝜓 → ( 𝜑 → 𝜒 ) ) ↔ ( ( 𝜓 ∧ 𝜑 ) → 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		impexp	⊢ ( ( ( 𝜑 ∧ 𝜓 ) → 𝜒 ) ↔ ( 𝜑 → ( 𝜓 → 𝜒 ) ) )
2		bi2.04	⊢ ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) ↔ ( 𝜓 → ( 𝜑 → 𝜒 ) ) )
3	1 2	pm3.2i	⊢ ( ( ( ( 𝜑 ∧ 𝜓 ) → 𝜒 ) ↔ ( 𝜑 → ( 𝜓 → 𝜒 ) ) ) ∧ ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) ↔ ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) )
4		impexp	⊢ ( ( ( 𝜓 ∧ 𝜑 ) → 𝜒 ) ↔ ( 𝜓 → ( 𝜑 → 𝜒 ) ) )
5	4	bicomi	⊢ ( ( 𝜓 → ( 𝜑 → 𝜒 ) ) ↔ ( ( 𝜓 ∧ 𝜑 ) → 𝜒 ) )
6	3 5	pm3.2i	⊢ ( ( ( ( ( 𝜑 ∧ 𝜓 ) → 𝜒 ) ↔ ( 𝜑 → ( 𝜓 → 𝜒 ) ) ) ∧ ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) ↔ ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ∧ ( ( 𝜓 → ( 𝜑 → 𝜒 ) ) ↔ ( ( 𝜓 ∧ 𝜑 ) → 𝜒 ) ) )