Theorem syl8 76

Description: A syllogism rule of inference. The second premise is used to replace the consequent of the first premise. (Contributed by NM, 1-Aug-1994.) (Proof shortened by Wolf Lammen, 3-Aug-2012.)

Hypotheses

Ref	Expression
syl8.1	⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))
syl8.2	⊢ (𝜃 → 𝜏)

Assertion

Ref	Expression
syl8	⊢ (𝜑 → (𝜓 → (𝜒 → 𝜏)))

Proof of Theorem syl8

Step	Hyp	Ref	Expression
1		syl8.1	. 2 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))
2		syl8.2	. . 3 ⊢ (𝜃 → 𝜏)
3	2	a1i 11	. 2 ⊢ (𝜑 → (𝜃 → 𝜏))
4	1, 3	syl6d 75	1 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜏)))

Syntax hints:   → wi 4

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

This theorem is referenced by:  a1ddd  80  com45  97  syl8ib  256  mo4  2559  ssorduni  7735  tz7.49  8390  nneneq  9147  dfac2b  10060  qreccl  12904  dvdsaddre2b  16253  cmpsub  23320  fclsopni  23935  nocvxminlem  27723  sumdmdlem2  32398  umgr2cycllem  35120  idinside  36065  axc11n11r  36664  isbasisrelowllem1  37336  isbasisrelowllem2  37337  dmqseqim2  38642  disjlem17  38784  prtlem15  38861  prtlem17  38862  ee3bir  44486  ee233  44502  onfrALTlem2  44529  ee223  44617  ee33VD  44861  ormkglobd  46866  rngccatidALTV  48253  ringccatidALTV  48287