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  9384  eqinfd  9413  updjud  9863  fvf1tp  13727  mreexexlemd  17585  mhmlem  18976  nn0gsumfz  19898  mdetunilem3  22534  mdetunilem9  22540  axtgupdim2  28451  axtgeucl  28452  wwlksnextprop  29892  measdivcst  34207  btwnouttr2  36003  btwnexch2  36004  cgrsub  36026  btwnconn1lem2  36069  btwnconn1lem5  36072  btwnconn1lem6  36073  segcon2  36086  btwnoutside  36106  broutsideof3  36107  outsideoftr  36110  outsideofeq  36111  lineelsb2  36129  relowlssretop  37344  lshpkrlem6  39101  reladdrsub  42366  onsupuni  43211  omabs2  43314  modelaxreplem2  44962  fmuldfeq  45574  stoweidlem5  45996  el0ldep  48448  ldepspr  48455