Theorem pm3.2an3 1131

Description: pm3.2 434 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		pm3.2 434	. . 3 ⊢ ((φ ∧ ψ) → (χ → ((φ ∧ ψ) ∧ χ)))
2	1	ex 423	. 2 ⊢ (φ → (ψ → (χ → ((φ ∧ ψ) ∧ χ))))
3		df-3an 936	. . 3 ⊢ ((φ ∧ ψ ∧ χ) ↔ ((φ ∧ ψ) ∧ χ))
4	3	bicomi 193	. 2 ⊢ (((φ ∧ ψ) ∧ χ) ↔ (φ ∧ ψ ∧ χ))
5	2, 4	syl8ib 222	1 ⊢ (φ → (ψ → (χ → (φ ∧ ψ ∧ χ))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 358   ∧ w3a 934

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-an 360  df-3an 936

This theorem is used by:  3exp  1150