Theorem 3anim123d 1259

Description: Deduction joining 3 implications to form implication of conjunctions. (Contributed by NM, 24-Feb-2005.)

Hypotheses

Ref	Expression
3anim123d.1	⊢ (φ → (ψ → χ))
3anim123d.2	⊢ (φ → (θ → τ))
3anim123d.3	⊢ (φ → (η → ζ))

Assertion

Ref	Expression
3anim123d	⊢ (φ → ((ψ ∧ θ ∧ η) → (χ ∧ τ ∧ ζ)))

Proof of Theorem 3anim123d

Step	Hyp	Ref	Expression
1		3anim123d.1	. . . 4 ⊢ (φ → (ψ → χ))
2		3anim123d.2	. . . 4 ⊢ (φ → (θ → τ))
3	1, 2	anim12d 546	. . 3 ⊢ (φ → ((ψ ∧ θ) → (χ ∧ τ)))
4		3anim123d.3	. . 3 ⊢ (φ → (η → ζ))
5	3, 4	anim12d 546	. 2 ⊢ (φ → (((ψ ∧ θ) ∧ η) → ((χ ∧ τ) ∧ ζ)))
6		df-3an 936	. 2 ⊢ ((ψ ∧ θ ∧ η) ↔ ((ψ ∧ θ) ∧ η))
7		df-3an 936	. 2 ⊢ ((χ ∧ τ ∧ ζ) ↔ ((χ ∧ τ) ∧ ζ))
8	5, 6, 7	3imtr4g 261	1 ⊢ (φ → ((ψ ∧ θ ∧ η) → (χ ∧ τ ∧ ζ)))

Syntax hints:   → wi 4   ∧ wa 358   ∧ w3a 934

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

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

This theorem is referenced by: (None)