Theorem dedth3h 2388

Description: Weak deduction theorem eliminating three hypotheses. See comments in dedth2h 2387.

Hypotheses

Ref	Expression
dedth3h.1	|- (A = if(ph, A, D) -> (th <-> ta))
dedth3h.2	|- (B = if(ps, B, R) -> (ta <-> et))
dedth3h.3	|- (C = if(ch, C, S) -> (et <-> ze))
dedth3h.4	|- ze

Assertion

Ref	Expression
dedth3h	|- ((ph /\ ps /\ ch) -> th)

Proof of Theorem dedth3h

Step	Hyp	Ref	Expression
1		dedth3h.1	. . . 4 |- (A = if(ph, A, D) -> (th <-> ta))
2	1	imbi2d 612	. . 3 |- (A = if(ph, A, D) -> (((ps /\ ch) -> th) <-> ((ps /\ ch) -> ta)))
3		dedth3h.2	. . . 4 |- (B = if(ps, B, R) -> (ta <-> et))
4		dedth3h.3	. . . 4 |- (C = if(ch, C, S) -> (et <-> ze))
5		dedth3h.4	. . . 4 |- ze
6	3, 4, 5	dedth2h 2387	. . 3 |- ((ps /\ ch) -> ta)
7	2, 6	dedth 2383	. 2 |- (ph -> ((ps /\ ch) -> th))
8	7	3impib 831	1 |- ((ph /\ ps /\ ch) -> th)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 775   = wceq 956  ifcif 2361

This theorem is referenced by:  addcant 5352  subaddt 5375  ltadd1t 5623  leadd1t 5625  ltsubaddt 5627  lesubaddt 5629  mulcant2 5688  mulcant2OLD 5689  divmult 5707  divdirt 5750  divdirtOLD 5751  div11t 5765  ltmul1t 5830  ltdiv1t 5849  ltdiv1tOLD 5850  ltmuldivt 5863  ltmuldivtOLD 5864  icoshftf1olem 6410  icoun 6413  faclbnd4lem2 6949  efcnlem4 7422  ipdiri 8489  efifolem3 8724  hvaddcant 8937  hvsubaddt 8944  norm3dift 9017  omlsi 9245  shlubt 9354  chjasst 9456  ledit 9463  spansncvt 9598  pjcjt2 9637  pjopytht 9665  hoaddasst 9708  hocsubdirt 9711  hoddit 9915

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-if 2362

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