Theorem 3jcad 1127

Description: Deduction conjoining the consequents of three implications. (Contributed by NM, 25-Sep-2005.)

Hypotheses

Ref	Expression
3jcad.1	⊢ (𝜑 → (𝜓 → 𝜒))
3jcad.2	⊢ (𝜑 → (𝜓 → 𝜃))
3jcad.3	⊢ (𝜑 → (𝜓 → 𝜏))

Assertion

Ref	Expression
3jcad	⊢ (𝜑 → (𝜓 → (𝜒 ∧ 𝜃 ∧ 𝜏)))

Proof of Theorem 3jcad

Step	Hyp	Ref	Expression
1		3jcad.1	. . . 4 ⊢ (𝜑 → (𝜓 → 𝜒))
2	1	imp 406	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
3		3jcad.2	. . . 4 ⊢ (𝜑 → (𝜓 → 𝜃))
4	3	imp 406	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜃)
5		3jcad.3	. . . 4 ⊢ (𝜑 → (𝜓 → 𝜏))
6	5	imp 406	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜏)
7	2, 4, 6	3jca 1126	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ∧ 𝜃 ∧ 𝜏))
8	7	ex 412	1 ⊢ (𝜑 → (𝜓 → (𝜒 ∧ 𝜃 ∧ 𝜏)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ 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 396  df-3an 1087

This theorem is referenced by:  onfununi  8143  uzm1  12545  ixxssixx  13022  iccid  13053  iccsplit  13146  fzen  13202  lmodprop2d  20100  fbun  22899  hausflim  23040  icoopnst  24008  iocopnst  24009  abelth  25505  usgr2pth  28033  shsvs  29586  cnlnssadj  30343  fnrelpredd  32961  cvmlift2lem10  33174  endofsegid  34314  elicc3  34433  areacirclem1  35792  islvol2aN  37533  alrim3con13v  42042  bgoldbtbndlem4  45148