Theorem syl6c 71

Description: Inference combining syl6 36 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 36	. 2 ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜏)))
5	1, 4	mpdd 44	1 ⊢ (𝜑 → (𝜓 → 𝜏))

This proof depends on syntax axioms:   → wi 4

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7

This theorem is used by:  syl6ci  72  syldd  73  impbidd  213  pm5.21ndd  382  jcad  522  a2and  859  zorn2lem6  10606  sqreulem  15556  ontopbas  37394  ontgval  37397  ordtoplem  37401  ordcmp  37413  fvineqsneu  38526  jaodd  43452  ee33  45695  sb5ALT  45699  tratrb  45710  onfrALTlem2  45720  onfrALT  45723  ax6e2ndeq  45733  ee22an  45847  sspwtrALT  45995  sspwtrALT2  45996  trintALT  46054