Theorem mp3and 1466

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 1128	. 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 1086

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 1088

This theorem is referenced by:  eqsupd  9469  eqinfd  9498  updjud  9948  fvf1tp  13806  mreexexlemd  17656  mhmlem  19045  nn0gsumfz  19965  mdetunilem3  22552  mdetunilem9  22558  axtgupdim2  28450  axtgeucl  28451  wwlksnextprop  29894  measdivcst  34255  btwnouttr2  36040  btwnexch2  36041  cgrsub  36063  btwnconn1lem2  36106  btwnconn1lem5  36109  btwnconn1lem6  36110  segcon2  36123  btwnoutside  36143  broutsideof3  36144  outsideoftr  36147  outsideofeq  36148  lineelsb2  36166  relowlssretop  37381  lshpkrlem6  39133  reladdrsub  42428  onsupuni  43253  omabs2  43356  modelaxreplem2  45004  fmuldfeq  45612  stoweidlem5  46034  el0ldep  48442  ldepspr  48449