Theorem 3jcad 1129

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 407	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
3		3jcad.2	. . . 4 ⊢ (𝜑 → (𝜓 → 𝜃))
4	3	imp 407	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜃)
5		3jcad.3	. . . 4 ⊢ (𝜑 → (𝜓 → 𝜏))
6	5	imp 407	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜏)
7	2, 4, 6	3jca 1128	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ∧ 𝜃 ∧ 𝜏))
8	7	ex 413	1 ⊢ (𝜑 → (𝜓 → (𝜒 ∧ 𝜃 ∧ 𝜏)))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087

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 1089

This theorem is referenced by:  onfununi  8340  uzm1  12859  ixxssixx  13337  iccid  13368  iccsplit  13461  fzen  13517  lmodprop2d  20533  fbun  23343  hausflim  23484  icoopnst  24454  iocopnst  24455  abelth  25952  usgr2pth  29018  shsvs  30571  cnlnssadj  31328  fnrelpredd  34087  cvmlift2lem10  34298  endofsegid  35052  elicc3  35197  areacirclem1  36571  islvol2aN  38458  alrim3con13v  43284  bgoldbtbndlem4  46466