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 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  9497  eqinfd  9525  updjud  9974  fvf1tp  13829  mreexexlemd  17687  mhmlem  19080  nn0gsumfz  20002  mdetunilem3  22620  mdetunilem9  22626  axtgupdim2  28479  axtgeucl  28480  wwlksnextprop  29932  measdivcst  34225  btwnouttr2  36023  btwnexch2  36024  cgrsub  36046  btwnconn1lem2  36089  btwnconn1lem5  36092  btwnconn1lem6  36093  segcon2  36106  btwnoutside  36126  broutsideof3  36127  outsideoftr  36130  outsideofeq  36131  lineelsb2  36149  relowlssretop  37364  lshpkrlem6  39116  reladdrsub  42415  onsupuni  43241  omabs2  43345  modelaxreplem2  44996  fmuldfeq  45598  stoweidlem5  46020  el0ldep  48383  ldepspr  48390