Theorem syldanl 604

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 416	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
3	2	imdistani 572	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝜑 ∧ 𝜒))
4		syldanl.2	. 2 ⊢ (((𝜑 ∧ 𝜒) ∧ 𝜃) → 𝜏)
5	3, 4	sylan 583	1 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜃) → 𝜏)

Syntax hints:   → wi 4   ∧ wa 399

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 210  df-an 400

This theorem is referenced by:  sylanl2  680  oen0  8195  oeordsuc  8203  erth  8321  lo1bdd2  14873  grplmulf1o  18165  grplactcnv  18194  trust  22835  efrlim  25555  fedgmullem2  31114  submateq  31162  heibor1lem  35247  idlnegcl  35460  igenmin  35502  eqvrelth  36006  binomcxplemnotnn0  41060  vonioolem1  43319  vonicclem1  43322  smfsuplem1  43442  smflimsuplem4  43454