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 411	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
3		3jcad.2	. . . 4 ⊢ (𝜑 → (𝜓 → 𝜃))
4	3	imp 411	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜃)
5		3jcad.3	. . . 4 ⊢ (𝜑 → (𝜓 → 𝜏))
6	5	imp 411	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜏)
7	2, 4, 6	3jca 1146	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ∧ 𝜃 ∧ 𝜏))
8	7	ex 417	1 ⊢ (𝜑 → (𝜓 → (𝜒 ∧ 𝜃 ∧ 𝜏)))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1103

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 210  df-an 401  df-3an 1105

This theorem is referenced by:  onfununi  8324  uzm1  12891  ixxssixx  13381  iccid  13412  iccsplit  13507  fzen  13564  lmodprop2d  21045  fbun  23997  hausflim  24138  icoopnst  25098  iocopnst  25099  abelth  26604  usgr2pth  30113  shsvs  31675  cnlnssadj  32432  fnrelpredd  35482  trssfir1om  35507  trssfir1omregs  35549  cvmlift2lem10  35804  endofsegid  36577  elicc3  36828  areacirclem1  38359  islvol2aN  40366  alrim3con13v  45242  ormkglobd  47591  bgoldbtbndlem4  48573