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  1407  disjiun  4024  tfrlem1  6361  tfrcl  6417  mkvprop  7217  ccfunen  7324  caucvgprprlemval  7748  suplocsrlem  7868  peano5uzti  9425  seqf1oglem2  10591  zfz1iso  10912  lcmneg  12212  prmind2  12258  pcfac  12488  cnmpt12  14455  cnmpt22  14462  limccnp2lem  14830  2sqlem6  15207  2sqlem8  15210  sbthom  15516