Theorem syl6c 70

Description: Inference combining syl6 35 with contraction. (Contributed by Alan Sare, 2-May-2011.)

Hypotheses

Ref	Expression
syl6c.1	⊢ (𝜑 → (𝜓 → 𝜒))
syl6c.2	⊢ (𝜑 → (𝜓 → 𝜃))
syl6c.3	⊢ (𝜒 → (𝜃 → 𝜏))

Assertion

Ref	Expression
syl6c	⊢ (𝜑 → (𝜓 → 𝜏))

Proof of Theorem syl6c

Step	Hyp	Ref	Expression
1		syl6c.2	. 2 ⊢ (𝜑 → (𝜓 → 𝜃))
2		syl6c.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
3		syl6c.3	. . 3 ⊢ (𝜒 → (𝜃 → 𝜏))
4	2, 3	syl6 35	. 2 ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜏)))
5	1, 4	mpdd 43	1 ⊢ (𝜑 → (𝜓 → 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:  syl6ci  71  syldd  72  impbidd  212  pm5.21ndd  383  jcad  515  a2and  841  zorn2lem6  9925  sqreulem  14721  ontopbas  33778  ontgval  33781  ordtoplem  33785  ordcmp  33797  fvineqsneu  34694  jaodd  39107  ee33  40862  sb5ALT  40866  tratrb  40877  onfrALTlem2  40887  onfrALT  40890  ax6e2ndeq  40900  ee22an  41014  sspwtrALT  41163  sspwtrALT2  41164  trintALT  41222