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 1128	. 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 1086

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 1088

This theorem is referenced by:  eqsupd  9347  eqinfd  9376  updjud  9830  fvf1tp  13693  mreexexlemd  17550  mhmlem  18941  nn0gsumfz  19863  mdetunilem3  22499  mdetunilem9  22505  axtgupdim2  28420  axtgeucl  28421  wwlksnextprop  29861  measdivcst  34207  btwnouttr2  36016  btwnexch2  36017  cgrsub  36039  btwnconn1lem2  36082  btwnconn1lem5  36085  btwnconn1lem6  36086  segcon2  36099  btwnoutside  36119  broutsideof3  36120  outsideoftr  36123  outsideofeq  36124  lineelsb2  36142  relowlssretop  37357  lshpkrlem6  39114  reladdrsub  42378  onsupuni  43222  omabs2  43325  modelaxreplem2  44973  fmuldfeq  45584  stoweidlem5  46006  el0ldep  48471  ldepspr  48478