Theorem mp3and 1493

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 1146	. 2 ⊢ (𝜑 → (𝜓 ∧ 𝜒 ∧ 𝜃))
5		mp3and.4	. 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏))
6	4, 5	mpd 16	1 ⊢ (𝜑 → 𝜏)

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105

This theorem is used by:  eqsupd  9427  eqinfd  9456  updjud  9986  fvf1tp  13897  mreexexlemd  17779  mhmlem  19233  nn0gsumfz  20159  mdetunilem3  22890  mdetunilem9  22896  axtgupdim2  28866  axtgeucl  28867  wwlksnextprop  30434  measdivcst  34790  btwnouttr2  36709  btwnexch2  36710  cgrsub  36732  btwnconn1lem2  36775  btwnconn1lem5  36778  btwnconn1lem6  36779  segcon2  36792  btwnoutside  36812  broutsideof3  36813  outsideoftr  36816  outsideofeq  36817  lineelsb2  36835  relowlssretop  38206  lshpkrlem6  40092  reladdrsub  43364  onsupuni  44174  omabs2  44277  modelaxreplem2  45906  fmuldfeq  46517  stoweidlem5  46937  el0ldep  49500  ldepspr  49507