Theorem 3jcad 1147

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 412	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
3		3jcad.2	. . . 4 ⊢ (𝜑 → (𝜓 → 𝜃))
4	3	imp 412	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜃)
5		3jcad.3	. . . 4 ⊢ (𝜑 → (𝜓 → 𝜏))
6	5	imp 412	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜏)
7	2, 4, 6	3jca 1146	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ∧ 𝜃 ∧ 𝜏))
8	7	ex 418	1 ⊢ (𝜑 → (𝜓 → (𝜒 ∧ 𝜃 ∧ 𝜏)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105

This theorem is used by:  onfununi  8330  uzm1  12921  ixxssixx  13412  iccid  13443  iccsplit  13538  fzen  13595  lmodprop2d  21108  fbun  24066  hausflim  24207  icoopnst  25167  iocopnst  25168  abelth  26677  usgr2pth  30229  shsvs  31804  cnlnssadj  32561  fnrelpredd  35596  trssfir1om  35621  trssfir1omregs  35662  cvmlift2lem10  35891  endofsegid  36665  elicc3  36936  areacirclem1  38457  islvol2aN  40465  alrim3con13v  45356  ormkglobd  47705  bgoldbtbndlem4  48724