Theorem syldanl 614

Description: A syllogism deduction with conjoined antecedents. (Contributed by Jeff Madsen, 20-Jun-2011.)

Hypotheses

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

Assertion

Ref	Expression
syldanl	⊢ (((𝜑 ∧ 𝜓) ∧ 𝜃) → 𝜏)

Proof of Theorem syldanl

Step	Hyp	Ref	Expression
1		syldanl.1	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
2	1	ex 418	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
3	2	imdistani 579	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝜑 ∧ 𝜒))
4		syldanl.2	. 2 ⊢ (((𝜑 ∧ 𝜒) ∧ 𝜃) → 𝜏)
5	3, 4	sylan 592	1 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜃) → 𝜏)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402

This theorem is used by:  sylanl2  694  oen0  8581  oeordsuc  8589  erth  8758  phplem2  9199  lo1bdd2  15601  grplmulf1o  19110  grplactcnv  19140  trust  24423  efrlim  27171  suppgsumssiun  33423  evlextv  33963  fedgmullem2  34051  submateq  34230  heibor1lem  38501  idlnegcl  38714  igenmin  38756  eqvrelth  39385  sticksstones22  42976  binomcxplemnotnn0  45107  vonioolem1  47435  vonicclem1  47438  smfsuplem1  47566  smflimsuplem4  47578