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 1130	. 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 1089

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 210  df-an 400  df-3an 1091

This theorem is referenced by:  eqsupd  9073  eqinfd  9101  updjud  9550  mreexexlemd  17147  mhmlem  18483  nn0gsumfz  19369  mdetunilem3  21511  mdetunilem9  21517  axtgupdim2  26562  axtgeucl  26563  wwlksnextprop  27996  measdivcst  31904  btwnouttr2  34061  btwnexch2  34062  cgrsub  34084  btwnconn1lem2  34127  btwnconn1lem5  34130  btwnconn1lem6  34131  segcon2  34144  btwnoutside  34164  broutsideof3  34165  outsideoftr  34168  outsideofeq  34169  lineelsb2  34187  relowlssretop  35271  lshpkrlem6  36866  reladdrsub  40076  fmuldfeq  42799  stoweidlem5  43221  el0ldep  45480  ldepspr  45487