Theorem mp3an12i 1341

Description: mp3an 1337 with antecedents in standard conjunction form and with one hypothesis an implication. (Contributed by Alan Sare, 28-Aug-2016.)

Hypotheses

Ref	Expression
mp3an12i.1	⊢ 𝜑
mp3an12i.2	⊢ 𝜓
mp3an12i.3	⊢ (𝜒 → 𝜃)
mp3an12i.4	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏)

Assertion

Ref	Expression
mp3an12i	⊢ (𝜒 → 𝜏)

Proof of Theorem mp3an12i

Step	Hyp	Ref	Expression
1		mp3an12i.3	. 2 ⊢ (𝜒 → 𝜃)
2		mp3an12i.1	. . 3 ⊢ 𝜑
3		mp3an12i.2	. . 3 ⊢ 𝜓
4		mp3an12i.4	. . 3 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏)
5	2, 3, 4	mp3an12 1327	. 2 ⊢ (𝜃 → 𝜏)
6	1, 5	syl 14	1 ⊢ (𝜒 → 𝜏)

Syntax hints:   → wi 4   ∧ w3a 978

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117  df-3an 980

This theorem is referenced by:  map1  6814  suplocsrlempr  7808  geo2lim  11526  fprodge0  11647  fprodge1  11649  oddp1d2  11897  bezoutlema  12002  bezoutlemb  12003  pythagtriplem1  12267  exmidunben  12429  ismet  13929  isxmet  13930  coseq0negpitopi  14342  cosq34lt1  14356  cos02pilt1  14357  logdivlti  14387  lgseisenlem1  14535  m1lgs  14537