Theorem mp3and 1473

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 1135	. 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 1093

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 209  df-an 398  df-3an 1095

This theorem is referenced by:  eqsupd  9364  eqinfd  9393  updjud  9853  fvf1tp  13743  mreexexlemd  17605  mhmlem  19033  nn0gsumfz  19954  mdetunilem3  22601  mdetunilem9  22607  axtgupdim2  28561  axtgeucl  28562  wwlksnextprop  30002  measdivcst  34420  btwnouttr2  36265  btwnexch2  36266  cgrsub  36288  btwnconn1lem2  36331  btwnconn1lem5  36334  btwnconn1lem6  36335  segcon2  36348  btwnoutside  36368  broutsideof3  36369  outsideoftr  36372  outsideofeq  36373  lineelsb2  36391  relowlssretop  37740  lshpkrlem6  39622  reladdrsub  42877  onsupuni  43689  omabs2  43792  modelaxreplem2  45438  fmuldfeq  46042  stoweidlem5  46462  el0ldep  48971  ldepspr  48978