Theorem syl6an 1483

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))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia3 108

This theorem is used by:  mapxpen  7148  prarloclem5  7867  ltsopr  7963  suplocsrlem  8175  nominpos  9545  ublbneg  10015  wrdsymb0  11339  ccats1pfxeqrex  11489  absle  11857  rexanre  11988  rexico  11989  climshftlemg  12070  serf0  12120  dvds1lem  12571  dvds2lem  12572  lmconst  15319  addcncntoplem  15664  bj-indind  16970