Theorem syl3an3 1170

Description: A syllogism inference. (Contributed by NM, 22-Aug-1995.)

Hypotheses

Ref	Expression
syl3an3.1	⊢ (𝜑 → 𝜃)
syl3an3.2	⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏)

Assertion

Ref	Expression
syl3an3	⊢ ((𝜓 ∧ 𝜒 ∧ 𝜑) → 𝜏)

Proof of Theorem syl3an3

Step	Hyp	Ref	Expression
1		syl3an3.1	. . 3 ⊢ (𝜑 → 𝜃)
2		syl3an3.2	. . . 4 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏)
3	2	3exp 1103	. . 3 ⊢ (𝜓 → (𝜒 → (𝜃 → 𝜏)))
4	1, 3	syl7 63	. 2 ⊢ (𝜓 → (𝜒 → (𝜑 → 𝜏)))
5	4	3imp 1098	1 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜑) → 𝜏)

Syntax hints:   → wi 4   ∧ w3a 885

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101

This theorem depends on definitions:  df-bi 110  df-3an 887

This theorem is referenced by:  syl3an3b  1173  syl3an3br  1176  vtoclgft  2604  ovmpt2x  5629  ovmpt2ga  5630  nnanq0  6556  apreim  7594  divassap  7669  ltmul2  7822  elfzo  9006  subcn2  9832  mulcn2  9833