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  2649  fvf1pr  7255  frxp  8069  poxp2  8086  mapen  9072  rex2dom  9156  fin1a2lem9  10321  coprmproddvdslem  16622  psss  18537  mgmidmo  18619  aannenlem1  26305  funtransport  36229  cgrxfr  36253  btwnxfr  36254  weiunpo  36663  bj-cbv3tb  37110