Theorem syld3an1 1266

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 1190	. . 3 ⊢ ((𝜃 ∧ 𝜓 ∧ 𝜒) → 𝜑)
3		syld3an1.2	. . . 4 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏)
4	3	3com13 1190	. . 3 ⊢ ((𝜃 ∧ 𝜓 ∧ 𝜑) → 𝜏)
5	2, 4	syld3an3 1265	. 2 ⊢ ((𝜃 ∧ 𝜓 ∧ 𝜒) → 𝜏)
6	5	3com13 1190	1 ⊢ ((𝜒 ∧ 𝜓 ∧ 𝜃) → 𝜏)

Syntax hints:   → wi 4   ∧ w3a 963

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116  df-3an 965

This theorem is referenced by:  tfrcllembacc  6296  npncan  8079  nnpcan  8081  ppncan  8100  muldivdirap  8563  div2negap  8591  ltmuldiv  8728  mulqmod0  10211