Theorem 3adant1l 1174

Description: Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.)

Hypothesis

Ref	Expression
3adant1l.1	⊢ ((φ ∧ ψ ∧ χ) → θ)

Assertion

Ref	Expression
3adant1l	⊢ (((τ ∧ φ) ∧ ψ ∧ χ) → θ)

Proof of Theorem 3adant1l

Step	Hyp	Ref	Expression
1		3adant1l.1	. . . 4 ⊢ ((φ ∧ ψ ∧ χ) → θ)
2	1	3expb 1152	. . 3 ⊢ ((φ ∧ (ψ ∧ χ)) → θ)
3	2	adantll 694	. 2 ⊢ (((τ ∧ φ) ∧ (ψ ∧ χ)) → θ)
4	3	3impb 1147	1 ⊢ (((τ ∧ φ) ∧ ψ ∧ χ) → θ)

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:  3adant2l  1176  3adant3l  1178