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Metamath Proof Explorer

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 ) ) )