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  10398  sqreulem  15273  ontopbas  36479  ontgval  36482  ordtoplem  36486  ordcmp  36498  fvineqsneu  37462  jaodd  42307  ee33  44619  sb5ALT  44623  tratrb  44634  onfrALTlem2  44644  onfrALT  44647  ax6e2ndeq  44657  ee22an  44771  sspwtrALT  44919  sspwtrALT2  44920  trintALT  44978