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)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem is used by:  adantlll  484  adantllr  485  adantl3r  516  isocnv  6020  mapxpen  7153  nqnq0pi  7812  nqpnq0nq  7827  addnqprl  7903  addnqpru  7904  pcqmul  13112  infpnlem1  13168  setsn0fun  13448  dvmptfsum  15963  chtqub  16308  usgr2edg  16666  usgr2edg1  16668