Theorem pm3.2an3 1171

Description: pm3.2 138 for a triple conjunction. (Contributed by Alan Sare, 24-Oct-2011.)

Assertion

Ref	Expression
pm3.2an3	⊢ (𝜑 → (𝜓 → (𝜒 → (𝜑 ∧ 𝜓 ∧ 𝜒))))

Proof of Theorem pm3.2an3

Step	Hyp	Ref	Expression
1		pm3.2 138	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 → ((𝜑 ∧ 𝜓) ∧ 𝜒)))
2	1	ex 114	. 2 ⊢ (𝜑 → (𝜓 → (𝜒 → ((𝜑 ∧ 𝜓) ∧ 𝜒))))
3		df-3an 975	. . 3 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒))
4	3	bicomi 131	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ (𝜑 ∧ 𝜓 ∧ 𝜒))
5	2, 4	syl8ib 165	1 ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜑 ∧ 𝜓 ∧ 𝜒))))

Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 973

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116  df-3an 975

This theorem is referenced by:  3exp  1197