Theorem syldanl 603

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 415	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
3	2	imdistani 571	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝜑 ∧ 𝜒))
4		syldanl.2	. 2 ⊢ (((𝜑 ∧ 𝜒) ∧ 𝜃) → 𝜏)
5	3, 4	sylan 582	1 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜃) → 𝜏)

Syntax hints:   → wi 4   ∧ wa 398

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 209  df-an 399

This theorem is referenced by:  sylanl2  679  oen0  8211  oeordsuc  8219  erth  8337  lo1bdd2  14880  grplmulf1o  18172  grplactcnv  18201  trust  22837  efrlim  25546  fedgmullem2  31026  submateq  31074  heibor1lem  35086  idlnegcl  35299  igenmin  35341  eqvrelth  35845  binomcxplemnotnn0  40686  vonioolem1  42961  vonicclem1  42964  smfsuplem1  43084  smflimsuplem4  43096