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  10550  sqreulem  15494  ontopbas  37138  ontgval  37141  ordtoplem  37145  ordcmp  37157  fvineqsneu  38254  jaodd  43180  ee33  45448  sb5ALT  45452  tratrb  45463  onfrALTlem2  45473  onfrALT  45476  ax6e2ndeq  45486  ee22an  45600  sspwtrALT  45748  sspwtrALT2  45749  trintALT  45807