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  7029  prarloclem5  7713  ltsopr  7809  suplocsrlem  8021  nominpos  9375  ublbneg  9840  wrdsymb0  11139  ccats1pfxeqrex  11289  absle  11643  rexanre  11774  rexico  11775  climshftlemg  11856  serf0  11906  dvds1lem  12356  dvds2lem  12357  lmconst  14933  addcncntoplem  15278  bj-indind  16477