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)

Syntax hints:   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7

This theorem is referenced by:  bilukdc  1418  disjiun  4057  tfrlem1  6424  tfrcl  6480  mkvprop  7293  ccfunen  7418  caucvgprprlemval  7843  suplocsrlem  7963  peano5uzti  9523  seqf1oglem2  10709  zfz1iso  11030  wrd2ind  11221  lcmneg  12562  prmind2  12608  pcfac  12839  cnmpt12  14926  cnmpt22  14933  limccnp2lem  15315  2sqlem6  15764  2sqlem8  15767  gropd  15813  grstructd2dom  15814  sbthom  16305