Theorem mp3and 1463

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 1127	. 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 206  df-an 397  df-3an 1088

This theorem is referenced by:  eqsupd  9216  eqinfd  9244  updjud  9692  mreexexlemd  17353  mhmlem  18695  nn0gsumfz  19585  mdetunilem3  21763  mdetunilem9  21769  axtgupdim2  26832  axtgeucl  26833  wwlksnextprop  28277  measdivcst  32192  btwnouttr2  34324  btwnexch2  34325  cgrsub  34347  btwnconn1lem2  34390  btwnconn1lem5  34393  btwnconn1lem6  34394  segcon2  34407  btwnoutside  34427  broutsideof3  34428  outsideoftr  34431  outsideofeq  34432  lineelsb2  34450  relowlssretop  35534  lshpkrlem6  37129  reladdrsub  40368  fmuldfeq  43124  stoweidlem5  43546  el0ldep  45807  ldepspr  45814