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  9367  eqinfd  9396  updjud  9855  fvf1tp  13745  mreexexlemd  17607  mhmlem  19035  nn0gsumfz  19956  mdetunilem3  22595  mdetunilem9  22601  axtgupdim2  28559  axtgeucl  28560  wwlksnextprop  30001  measdivcst  34390  btwnouttr2  36226  btwnexch2  36227  cgrsub  36249  btwnconn1lem2  36292  btwnconn1lem5  36295  btwnconn1lem6  36296  segcon2  36309  btwnoutside  36329  broutsideof3  36330  outsideoftr  36333  outsideofeq  36334  lineelsb2  36352  relowlssretop  37701  lshpkrlem6  39583  reladdrsub  42839  onsupuni  43683  omabs2  43786  modelaxreplem2  45432  fmuldfeq  46039  stoweidlem5  46459  el0ldep  48962  ldepspr  48969