Theorem syl3an2 1283

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 1204	. . 3 ⊢ (𝜓 → (𝜒 → (𝜃 → 𝜏)))
4	1, 3	syl5 32	. 2 ⊢ (𝜓 → (𝜑 → (𝜃 → 𝜏)))
5	4	3imp 1195	1 ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜃) → 𝜏)

Syntax hints:   → wi 4   ∧ w3a 980

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 982

This theorem is referenced by:  syl3an2b  1286  syl3an2br  1289  syl3anl2  1298  nndi  6506  nnmass  6507  prarloclemarch2  7443  1idprl  7614  1idpru  7615  recexprlem1ssl  7657  recexprlem1ssu  7658  msqge0  8598  mulge0  8601  divsubdirap  8690  divdiv32ap  8702  peano2uz  9608  fzoshftral  10263  expdivap  10597  bcval5  10770  redivap  10910  imdivap  10917  absdiflt  11128  absdifle  11129  retanclap  11757  tannegap  11763  lcmgcdeq  12110  isprm3  12145  prmdvdsexpb  12176  dvdsprmpweqnn  12363  mulgaddcomlem  13078  mulginvcom  13080  cnpf2  14144  blres  14371