Theorem syld3an1 1324

Description: A syllogism inference. (Contributed by NM, 7-Jul-2008.)

Hypotheses

Ref	Expression
syld3an1.1	⊢ ((𝜒 ∧ 𝜓 ∧ 𝜃) → 𝜑)
syld3an1.2	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏)

Assertion

Ref	Expression
syld3an1	⊢ ((𝜒 ∧ 𝜓 ∧ 𝜃) → 𝜏)

Proof of Theorem syld3an1

Step	Hyp	Ref	Expression
1		syld3an1.1	. . . 4 ⊢ ((𝜒 ∧ 𝜓 ∧ 𝜃) → 𝜑)
2	1	3com13 1239	. . 3 ⊢ ((𝜃 ∧ 𝜓 ∧ 𝜒) → 𝜑)
3		syld3an1.2	. . . 4 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏)
4	3	3com13 1239	. . 3 ⊢ ((𝜃 ∧ 𝜓 ∧ 𝜑) → 𝜏)
5	2, 4	syld3an3 1323	. 2 ⊢ ((𝜃 ∧ 𝜓 ∧ 𝜒) → 𝜏)
6	5	3com13 1239	1 ⊢ ((𝜒 ∧ 𝜓 ∧ 𝜃) → 𝜏)

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1009

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

This proof depends on definitions:  df-bi 117  df-3an 1011

This theorem is used by:  tfrcllembacc  6626  npncan  8549  nnpcan  8551  ppncan  8570  muldivdirap  9040  div2negap  9068  ltmuldiv  9207  mulqmod0  10782  gcdaddm  12780  zndvds  15068