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  210  pm5.21ndd  379  jcad  512  a2and  845  zorn2lem6  10454  sqreulem  15326  ontopbas  36416  ontgval  36419  ordtoplem  36423  ordcmp  36435  fvineqsneu  37399  jaodd  42196  ee33  44511  sb5ALT  44515  tratrb  44526  onfrALTlem2  44536  onfrALT  44539  ax6e2ndeq  44549  ee22an  44663  sspwtrALT  44811  sspwtrALT2  44812  trintALT  44870