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  9341  eqinfd  9370  updjud  9827  fvf1tp  13693  mreexexlemd  17550  mhmlem  18975  nn0gsumfz  19896  mdetunilem3  22529  mdetunilem9  22535  axtgupdim2  28449  axtgeucl  28450  wwlksnextprop  29890  measdivcst  34237  btwnouttr2  36066  btwnexch2  36067  cgrsub  36089  btwnconn1lem2  36132  btwnconn1lem5  36135  btwnconn1lem6  36136  segcon2  36149  btwnoutside  36169  broutsideof3  36170  outsideoftr  36173  outsideofeq  36174  lineelsb2  36192  relowlssretop  37407  lshpkrlem6  39213  reladdrsub  42477  onsupuni  43321  omabs2  43424  modelaxreplem2  45071  fmuldfeq  45682  stoweidlem5  46102  el0ldep  48566  ldepspr  48573