Theorem syl2ani 613

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 612	. 2 ⊢ (𝜓 → ((𝜒 ∧ 𝜂) → 𝜏))
5	1, 4	sylani 610	1 ⊢ (𝜓 → ((𝜑 ∧ 𝜂) → 𝜏))

Syntax hints:   → wi 4   ∧ wa 396

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 208  df-an 397

This theorem is referenced by:  2mo  2652  fvf1pr  7258  frxp  8073  poxp2  8090  mapen  9076  rex2dom  9160  fin1a2lem9  10328  coprmproddvdslem  16629  psss  18544  mgmidmo  18626  aannenlem1  26319  funtransport  36266  cgrxfr  36290  btwnxfr  36291  weiunpo  36700  bj-cbv3tb  37147