Theorem syld3an1 1319

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 1234	. . 3 ⊢ ((𝜃 ∧ 𝜓 ∧ 𝜒) → 𝜑)
3		syld3an1.2	. . . 4 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏)
4	3	3com13 1234	. . 3 ⊢ ((𝜃 ∧ 𝜓 ∧ 𝜑) → 𝜏)
5	2, 4	syld3an3 1318	. 2 ⊢ ((𝜃 ∧ 𝜓 ∧ 𝜒) → 𝜏)
6	5	3com13 1234	1 ⊢ ((𝜒 ∧ 𝜓 ∧ 𝜃) → 𝜏)

Syntax hints:   → wi 4   ∧ w3a 1004

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 depends on definitions:  df-bi 117  df-3an 1006

This theorem is referenced by:  tfrcllembacc  6520  npncan  8399  nnpcan  8401  ppncan  8420  muldivdirap  8886  div2negap  8914  ltmuldiv  9053  mulqmod0  10591  gcdaddm  12554  zndvds  14662