Theorem syl3an2 1312

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 1233	. . 3 ⊢ (𝜓 → (𝜒 → (𝜃 → 𝜏)))
4	1, 3	syl5 32	. 2 ⊢ (𝜓 → (𝜑 → (𝜃 → 𝜏)))
5	4	3imp 1224	1 ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜃) → 𝜏)

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1009

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This proof depends on definitions:  df-bi 117  df-3an 1011

This theorem is used by:  syl3an2b  1315  syl3an2br  1318  syl3anl2  1327  nndi  6764  nnmass  6765  prarloclemarch2  7792  1idprl  7963  1idpru  7964  recexprlem1ssl  8006  recexprlem1ssu  8007  msqge0  8952  mulge0  8955  divsubdirap  9046  divdiv32ap  9058  peano2uz  9998  fzoshftral  10673  expdivap  11047  bcval5  11222  ccats1val1g  11428  redivap  11660  imdivap  11667  absdiflt  11880  absdifle  11881  retanclap  12513  tannegap  12519  lcmgcdeq  12885  isprm3  12920  prmdvdsexpb  12952  dvdsprmpweqnn  13143  mulgaddcomlem  14006  mulginvcom  14008  cnpf2  15450  blres  15677