Theorem 3jaao 1249

Description: Inference conjoining and disjoining the antecedents of three implications. (Contributed by Jeff Hankins, 15-Aug-2009.) (Proof shortened by Andrew Salmon, 13-May-2011.)

Hypotheses

Ref	Expression
3jaao.1	⊢ (φ → (ψ → χ))
3jaao.2	⊢ (θ → (τ → χ))
3jaao.3	⊢ (η → (ζ → χ))

Assertion

Ref	Expression
3jaao	⊢ ((φ ∧ θ ∧ η) → ((ψ ∨ τ ∨ ζ) → χ))

Proof of Theorem 3jaao

Step	Hyp	Ref	Expression
1		3jaao.1	. . 3 ⊢ (φ → (ψ → χ))
2	1	3ad2ant1 976	. 2 ⊢ ((φ ∧ θ ∧ η) → (ψ → χ))
3		3jaao.2	. . 3 ⊢ (θ → (τ → χ))
4	3	3ad2ant2 977	. 2 ⊢ ((φ ∧ θ ∧ η) → (τ → χ))
5		3jaao.3	. . 3 ⊢ (η → (ζ → χ))
6	5	3ad2ant3 978	. 2 ⊢ ((φ ∧ θ ∧ η) → (ζ → χ))
7	2, 4, 6	3jaod 1246	1 ⊢ ((φ ∧ θ ∧ η) → ((ψ ∨ τ ∨ ζ) → χ))

This proof depends on syntax axioms:   → wi 4   ∨ w3o 933   ∧ 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-or 359  df-an 360  df-3or 935  df-3an 936

This theorem is used by: (None)