Theorem mp3and 1493

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 1146	. 2 ⊢ (𝜑 → (𝜓 ∧ 𝜒 ∧ 𝜃))
5		mp3and.4	. 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏))
6	4, 5	mpd 16	1 ⊢ (𝜑 → 𝜏)

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105

This theorem is used by:  eqsupd  9430  eqinfd  9459  updjud  9942  fvf1tp  13852  mreexexlemd  17736  mhmlem  19186  nn0gsumfz  20112  mdetunilem3  22837  mdetunilem9  22843  axtgupdim2  28810  axtgeucl  28811  wwlksnextprop  30366  measdivcst  34722  btwnouttr2  36589  btwnexch2  36590  cgrsub  36612  btwnconn1lem2  36655  btwnconn1lem5  36658  btwnconn1lem6  36659  segcon2  36672  btwnoutside  36692  broutsideof3  36693  outsideoftr  36696  outsideofeq  36697  lineelsb2  36715  relowlssretop  38104  lshpkrlem6  39975  reladdrsub  43247  onsupuni  44057  omabs2  44160  modelaxreplem2  45789  fmuldfeq  46400  stoweidlem5  46820  el0ldep  49383  ldepspr  49390