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  6546  nnmass  6547  prarloclemarch2  7489  1idprl  7660  1idpru  7661  recexprlem1ssl  7703  recexprlem1ssu  7704  msqge0  8646  mulge0  8649  divsubdirap  8738  divdiv32ap  8750  peano2uz  9660  fzoshftral  10317  expdivap  10685  bcval5  10858  redivap  11042  imdivap  11049  absdiflt  11260  absdifle  11261  retanclap  11890  tannegap  11896  lcmgcdeq  12262  isprm3  12297  prmdvdsexpb  12328  dvdsprmpweqnn  12516  mulgaddcomlem  13301  mulginvcom  13303  cnpf2  14469  blres  14696