Theorem syl6an 1476

Description: A syllogism deduction combined with conjoining antecedents. (Contributed by Alan Sare, 28-Oct-2011.)

Hypotheses

Ref	Expression
syl6an.1	⊢ (𝜑 → 𝜓)
syl6an.2	⊢ (𝜑 → (𝜒 → 𝜃))
syl6an.3	⊢ ((𝜓 ∧ 𝜃) → 𝜏)

Assertion

Ref	Expression
syl6an	⊢ (𝜑 → (𝜒 → 𝜏))

Proof of Theorem syl6an

Step	Hyp	Ref	Expression
1		syl6an.2	. . 3 ⊢ (𝜑 → (𝜒 → 𝜃))
2		syl6an.1	. . 3 ⊢ (𝜑 → 𝜓)
3	1, 2	jctild 316	. 2 ⊢ (𝜑 → (𝜒 → (𝜓 ∧ 𝜃)))
4		syl6an.3	. 2 ⊢ ((𝜓 ∧ 𝜃) → 𝜏)
5	3, 4	syl6 33	1 ⊢ (𝜑 → (𝜒 → 𝜏))

Syntax hints:   → wi 4   ∧ wa 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia3 108

This theorem is referenced by:  mapxpen  7022  prarloclem5  7703  ltsopr  7799  suplocsrlem  8011  nominpos  9365  ublbneg  9825  wrdsymb0  11122  ccats1pfxeqrex  11268  absle  11621  rexanre  11752  rexico  11753  climshftlemg  11834  serf0  11884  dvds1lem  12334  dvds2lem  12335  lmconst  14911  addcncntoplem  15256  bj-indind  16404