Theorem mp3an12i 1352

Description: mp3an 1348 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 1338	. 2 ⊢ (𝜃 → 𝜏)
6	1, 5	syl 14	1 ⊢ (𝜒 → 𝜏)

Syntax hints:   → wi 4   ∧ w3a 980

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 982

This theorem is referenced by:  map1  6880  exmidpw2en  6982  suplocsrlempr  7891  geo2lim  11698  fprodge0  11819  fprodge1  11821  3dvds  12046  oddp1d2  12072  bezoutlema  12191  bezoutlemb  12192  pythagtriplem1  12459  exmidunben  12668  psrelbas  14304  psraddcl  14308  psr0cl  14309  psr0lid  14310  psrnegcl  14311  psrlinv  14312  psrgrp  14313  psr1clfi  14316  ismet  14664  isxmet  14665  dvidrelem  15012  coseq0negpitopi  15156  cosq34lt1  15170  cos02pilt1  15171  logdivlti  15201  1sgm2ppw  15315  lgseisenlem1  15395  lgseisen  15399  lgsquad3  15409  m1lgs  15410  2omapen  15727