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Metamath Proof Explorer

Theorem 3impexpbicom

Description: Version of 3impexp where in addition the consequent is commuted. (Contributed by Alan Sare, 31-Dec-2011)

Ref Expression
Assertion 3impexpbicom ( ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ( 𝜃 ↔ 𝜏 ) ) ↔ ( 𝜑 → ( 𝜓 → ( 𝜒 → ( 𝜏 ↔ 𝜃 ) ) ) ) )

Proof

Step Hyp Ref Expression
1 bicom ⊢ ( ( 𝜃 ↔ 𝜏 ) ↔ ( 𝜏 ↔ 𝜃 ) )
2 imbi2 ⊢ ( ( ( 𝜃 ↔ 𝜏 ) ↔ ( 𝜏 ↔ 𝜃 ) ) → ( ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ( 𝜃 ↔ 𝜏 ) ) ↔ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ( 𝜏 ↔ 𝜃 ) ) ) )
3 2 biimpcd ⊢ ( ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ( 𝜃 ↔ 𝜏 ) ) → ( ( ( 𝜃 ↔ 𝜏 ) ↔ ( 𝜏 ↔ 𝜃 ) ) → ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ( 𝜏 ↔ 𝜃 ) ) ) )
4 1 3 mpi ⊢ ( ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ( 𝜃 ↔ 𝜏 ) ) → ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ( 𝜏 ↔ 𝜃 ) ) )
5 4 3expd ⊢ ( ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ( 𝜃 ↔ 𝜏 ) ) → ( 𝜑 → ( 𝜓 → ( 𝜒 → ( 𝜏 ↔ 𝜃 ) ) ) ) )
6 3impexp ⊢ ( ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ( 𝜏 ↔ 𝜃 ) ) ↔ ( 𝜑 → ( 𝜓 → ( 𝜒 → ( 𝜏 ↔ 𝜃 ) ) ) ) )
7 6 biimpri ⊢ ( ( 𝜑 → ( 𝜓 → ( 𝜒 → ( 𝜏 ↔ 𝜃 ) ) ) ) → ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ( 𝜏 ↔ 𝜃 ) ) )
8 7 1 imbitrrdi ⊢ ( ( 𝜑 → ( 𝜓 → ( 𝜒 → ( 𝜏 ↔ 𝜃 ) ) ) ) → ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ( 𝜃 ↔ 𝜏 ) ) )
9 5 8 impbii ⊢ ( ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ( 𝜃 ↔ 𝜏 ) ) ↔ ( 𝜑 → ( 𝜓 → ( 𝜒 → ( 𝜏 ↔ 𝜃 ) ) ) ) )