Theorem syl3an2 1208

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

Hypotheses

Ref	Expression
syl3an2.1	⊢ (𝜑 → 𝜒)
syl3an2.2	⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏)

Assertion

Ref	Expression
syl3an2	⊢ ((𝜓 ∧ 𝜑 ∧ 𝜃) → 𝜏)

Proof of Theorem syl3an2

Step	Hyp	Ref	Expression
1		syl3an2.1	. . 3 ⊢ (𝜑 → 𝜒)
2		syl3an2.2	. . . 4 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏)
3	2	3exp 1142	. . 3 ⊢ (𝜓 → (𝜒 → (𝜃 → 𝜏)))
4	1, 3	syl5 32	. 2 ⊢ (𝜓 → (𝜑 → (𝜃 → 𝜏)))
5	4	3imp 1137	1 ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜃) → 𝜏)

Syntax hints:   → wi 4   ∧ w3a 924

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106

This theorem depends on definitions:  df-bi 115  df-3an 926

This theorem is referenced by:  syl3an2b  1211  syl3an2br  1214  syl3anl2  1223  nndi  6229  nnmass  6230  prarloclemarch2  6957  1idprl  7128  1idpru  7129  recexprlem1ssl  7171  recexprlem1ssu  7172  msqge0  8069  mulge0  8072  divsubdirap  8149  divdiv32ap  8161  peano2uz  9040  fzoshftral  9614  expdivap  9971  ibcval5  10136  redivap  10273  imdivap  10280  absdiflt  10490  absdifle  10491  lcmgcdeq  11158  isprm3  11193  prmdvdsexpb  11221