Theorem mp3and 1464

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 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  9526  eqinfd  9554  updjud  10003  fvf1tp  13840  mreexexlemd  17702  mhmlem  19102  nn0gsumfz  20026  mdetunilem3  22641  mdetunilem9  22647  axtgupdim2  28497  axtgeucl  28498  wwlksnextprop  29945  measdivcst  34188  btwnouttr2  35986  btwnexch2  35987  cgrsub  36009  btwnconn1lem2  36052  btwnconn1lem5  36055  btwnconn1lem6  36056  segcon2  36069  btwnoutside  36089  broutsideof3  36090  outsideoftr  36093  outsideofeq  36094  lineelsb2  36112  relowlssretop  37329  lshpkrlem6  39071  reladdrsub  42361  onsupuni  43190  omabs2  43294  fmuldfeq  45504  stoweidlem5  45926  el0ldep  48195  ldepspr  48202