Theorem syld3an1 1364

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 1262	. . 3 ⊢ ((𝜃 ∧ 𝜓 ∧ 𝜒) → 𝜑)
3		syld3an1.2	. . . 4 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏)
4	3	3com13 1262	. . 3 ⊢ ((𝜃 ∧ 𝜓 ∧ 𝜑) → 𝜏)
5	2, 4	syld3an3 1363	. 2 ⊢ ((𝜃 ∧ 𝜓 ∧ 𝜒) → 𝜏)
6	5	3com13 1262	1 ⊢ ((𝜒 ∧ 𝜓 ∧ 𝜃) → 𝜏)

Syntax hints:   → wi 4   ∧ w3a 1031

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 196  df-an 385  df-3an 1033

This theorem is referenced by:  npncan  10181  nnpcan  10183  ppncan  10202  muldivdir  10599  div2neg  10627  ltmuldiv  10775  opfi1uzind  13138  opfi1uzindOLD  13144  sgrp2nmndlem4  17238  zndvds  19717  subdivcomb1  30864  wsuceq123  31004  atlrelat1  33626  cvlatcvr1  33646  dih11  35572  mullimc  38683  mullimcf  38690  icccncfext  38773  stoweidlem34  38927  stoweidlem49  38942  stoweidlem57  38950  sigarexp  39697  el0ldepsnzr  42050