Theorem syl3c 63

Description: A syllogism inference combined with contraction. (Contributed by Alan Sare, 7-Jul-2011.)

Hypotheses

Ref	Expression
syl3c.1	⊢ (𝜑 → 𝜓)
syl3c.2	⊢ (𝜑 → 𝜒)
syl3c.3	⊢ (𝜑 → 𝜃)
syl3c.4	⊢ (𝜓 → (𝜒 → (𝜃 → 𝜏)))

Assertion

Ref	Expression
syl3c	⊢ (𝜑 → 𝜏)

Proof of Theorem syl3c

Step	Hyp	Ref	Expression
1		syl3c.3	. 2 ⊢ (𝜑 → 𝜃)
2		syl3c.1	. . 3 ⊢ (𝜑 → 𝜓)
3		syl3c.2	. . 3 ⊢ (𝜑 → 𝜒)
4		syl3c.4	. . 3 ⊢ (𝜓 → (𝜒 → (𝜃 → 𝜏)))
5	2, 3, 4	sylc 62	. 2 ⊢ (𝜑 → (𝜃 → 𝜏))
6	1, 5	mpd 13	1 ⊢ (𝜑 → 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:  bilukdc  1445  disjiun  4125  tfrlem1  6579  tfrcl  6635  mkvprop  7498  ccfunen  7630  caucvgprprlemval  8055  suplocsrlem  8175  peano5uzti  9758  seqf1oglem2  10970  zfz1iso  11307  wrd2ind  11509  lcmneg  12868  prmind2  12914  pcfac  13149  cnmpt12  15437  cnmpt22  15444  limccnp2lem  15826  2sqlem6  16358  2sqlem8  16361  gropd  16407  grstructd2dom  16408  sbthom  17190