Theorem syl2ani 606

Description: A syllogism inference. (Contributed by NM, 3-Aug-1999.)

Hypotheses

Ref	Expression
syl2ani.1	⊢ (𝜑 → 𝜒)
syl2ani.2	⊢ (𝜂 → 𝜃)
syl2ani.3	⊢ (𝜓 → ((𝜒 ∧ 𝜃) → 𝜏))

Assertion

Ref	Expression
syl2ani	⊢ (𝜓 → ((𝜑 ∧ 𝜂) → 𝜏))

Proof of Theorem syl2ani

Step	Hyp	Ref	Expression
1		syl2ani.1	. 2 ⊢ (𝜑 → 𝜒)
2		syl2ani.2	. . 3 ⊢ (𝜂 → 𝜃)
3		syl2ani.3	. . 3 ⊢ (𝜓 → ((𝜒 ∧ 𝜃) → 𝜏))
4	2, 3	sylan2i 605	. 2 ⊢ (𝜓 → ((𝜒 ∧ 𝜂) → 𝜏))
5	1, 4	sylani 603	1 ⊢ (𝜓 → ((𝜑 ∧ 𝜂) → 𝜏))

Syntax hints:   → wi 4   ∧ wa 395

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 206  df-an 396

This theorem is referenced by:  2mo  2638  frxp  8109  poxp2  8126  mapen  9140  rex2dom  9245  fin1a2lem9  10402  coprmproddvdslem  16604  psss  18543  mgmidmo  18591  aannenlem1  26214  funtransport  35536  cgrxfr  35560  btwnxfr  35561  bj-cbv3tb  36173