Theorem mp3and 1462

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 1126	. 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 1085

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 206  df-an 395  df-3an 1087

This theorem is referenced by:  eqsupd  9456  eqinfd  9484  updjud  9933  mreexexlemd  17594  mhmlem  18983  nn0gsumfz  19895  mdetunilem3  22338  mdetunilem9  22344  axtgupdim2  27987  axtgeucl  27988  wwlksnextprop  29431  measdivcst  33518  btwnouttr2  35296  btwnexch2  35297  cgrsub  35319  btwnconn1lem2  35362  btwnconn1lem5  35365  btwnconn1lem6  35366  segcon2  35379  btwnoutside  35399  broutsideof3  35400  outsideoftr  35403  outsideofeq  35404  lineelsb2  35422  relowlssretop  36549  lshpkrlem6  38290  reladdrsub  41562  onsupuni  42282  omabs2  42386  fmuldfeq  44599  stoweidlem5  45021  el0ldep  47236  ldepspr  47243