Theorem sylanl1 406

Description: A syllogism inference. (Contributed by NM, 10-Mar-2005.)

Hypotheses

Ref	Expression
sylanl1.1	⊢ (𝜑 → 𝜓)
sylanl1.2	⊢ (((𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜏)

Assertion

Ref	Expression
sylanl1	⊢ (((𝜑 ∧ 𝜒) ∧ 𝜃) → 𝜏)

Proof of Theorem sylanl1

Step	Hyp	Ref	Expression
1		sylanl1.1	. . 3 ⊢ (𝜑 → 𝜓)
2	1	anim1i 340	. 2 ⊢ ((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜒))
3		sylanl1.2	. 2 ⊢ (((𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜏)
4	2, 3	sylan 283	1 ⊢ (((𝜑 ∧ 𝜒) ∧ 𝜃) → 𝜏)

Syntax hints:   → wi 4   ∧ wa 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem is referenced by:  adantlll  484  adantllr  485  adantl3r  516  isocnv  6011  mapxpen  7142  nqnq0pi  7799  nqpnq0nq  7814  addnqprl  7890  addnqpru  7891  pcqmul  13065  infpnlem1  13121  setsn0fun  13372  dvmptfsum  15809  usgr2edg  16432  usgr2edg1  16434