Theorem adantlll 313

Description: Deduction adding a conjunct to an antecedent.

Hypothesis

Ref	Expression
adantl2.1	⊢ (((φ ∧ ψ) ∧ χ) → θ)

Assertion

Ref	Expression
adantlll	⊢ ((((τ ∧ φ) ∧ ψ) ∧ χ) → θ)

Proof of Theorem adantlll

Step	Hyp	Ref	Expression
1		adantl2.1	. . . 4 ⊢ (((φ ∧ ψ) ∧ χ) → θ)
2	1	exp31 293	. . 3 ⊢ (φ → (ψ → (χ → θ)))
3	2	a1i 7	. 2 ⊢ (τ → (φ → (ψ → (χ → θ))))
4	3	imp41 286	1 ⊢ ((((τ ∧ φ) ∧ ψ) ∧ χ) → θ)

Syntax hints:   → wi 2   ∧ wa 196

This theorem is referenced by:  sbthlem8 3356

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-an 198

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