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  9314  eqinfd  9342  updjud  9791  mreexexlemd  17450  mhmlem  18791  nn0gsumfz  19680  mdetunilem3  21869  mdetunilem9  21875  axtgupdim2  27121  axtgeucl  27122  wwlksnextprop  28565  measdivcst  32490  btwnouttr2  34420  btwnexch2  34421  cgrsub  34443  btwnconn1lem2  34486  btwnconn1lem5  34489  btwnconn1lem6  34490  segcon2  34503  btwnoutside  34523  broutsideof3  34524  outsideoftr  34527  outsideofeq  34528  lineelsb2  34546  relowlssretop  35647  lshpkrlem6  37390  reladdrsub  40636  omabs2  41325  fmuldfeq  43468  stoweidlem5  43890  el0ldep  46166  ldepspr  46173