Theorem mp3and 1467

Description: A deduction based on modus ponens. (Contributed by Mario Carneiro, 24-Dec-2016.)

Hypotheses

Ref	Expression
mp3and.1	⊢ (𝜑 → 𝜓)
mp3and.2	⊢ (𝜑 → 𝜒)
mp3and.3	⊢ (𝜑 → 𝜃)
mp3and.4	⊢ (𝜑 → ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏))

Assertion

Ref	Expression
mp3and	⊢ (𝜑 → 𝜏)

Proof of Theorem mp3and

Step	Hyp	Ref	Expression
1		mp3and.1	. . 3 ⊢ (𝜑 → 𝜓)
2		mp3and.2	. . 3 ⊢ (𝜑 → 𝜒)
3		mp3and.3	. . 3 ⊢ (𝜑 → 𝜃)
4	1, 2, 3	3jca 1129	. 2 ⊢ (𝜑 → (𝜓 ∧ 𝜒 ∧ 𝜃))
5		mp3and.4	. 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏))
6	4, 5	mpd 15	1 ⊢ (𝜑 → 𝜏)

Syntax hints:   → wi 4   ∧ w3a 1087

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  df-3an 1089

This theorem is referenced by:  eqsupd  9374  eqinfd  9403  updjud  9860  fvf1tp  13723  mreexexlemd  17581  mhmlem  19009  nn0gsumfz  19930  mdetunilem3  22575  mdetunilem9  22581  axtgupdim2  28561  axtgeucl  28562  wwlksnextprop  30003  measdivcst  34408  btwnouttr2  36244  btwnexch2  36245  cgrsub  36267  btwnconn1lem2  36310  btwnconn1lem5  36313  btwnconn1lem6  36314  segcon2  36327  btwnoutside  36347  broutsideof3  36348  outsideoftr  36351  outsideofeq  36352  lineelsb2  36370  relowlssretop  37645  lshpkrlem6  39520  reladdrsub  42784  onsupuni  43615  omabs2  43718  modelaxreplem2  45364  fmuldfeq  45972  stoweidlem5  46392  el0ldep  48855  ldepspr  48862