Theorem syl2ani 608

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 607	. 2 ⊢ (𝜓 → ((𝜒 ∧ 𝜂) → 𝜏))
5	1, 4	sylani 605	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 207  df-an 396

This theorem is referenced by:  2mo  2648  fvf1pr  7262  frxp  8076  poxp2  8093  mapen  9079  rex2dom  9163  fin1a2lem9  10330  coprmproddvdslem  16631  psss  18546  mgmidmo  18628  aannenlem1  26294  funtransport  36213  cgrxfr  36237  btwnxfr  36238  weiunpo  36647  bj-cbv3tb  37094