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  9359  eqinfd  9388  updjud  9847  fvf1tp  13737  mreexexlemd  17599  mhmlem  19027  nn0gsumfz  19948  mdetunilem3  22567  mdetunilem9  22573  axtgupdim2  28527  axtgeucl  28528  wwlksnextprop  29968  measdivcst  34356  btwnouttr2  36192  btwnexch2  36193  cgrsub  36215  btwnconn1lem2  36258  btwnconn1lem5  36261  btwnconn1lem6  36262  segcon2  36275  btwnoutside  36295  broutsideof3  36296  outsideoftr  36299  outsideofeq  36300  lineelsb2  36318  relowlssretop  37667  lshpkrlem6  39549  reladdrsub  42805  onsupuni  43645  omabs2  43748  modelaxreplem2  45394  fmuldfeq  46001  stoweidlem5  46421  el0ldep  48930  ldepspr  48937