Theorem dedth4h 1789

Description: Weak deduction theorem eliminating four hypotheses.

Hypotheses

Ref	Expression
dedth4h.1	|- (A = if(ph, A, R) -> (ta <-> et))
dedth4h.2	|- (B = if(ps, B, S) -> (et <-> ze))
dedth4h.3	|- (C = if(ch, C, F) -> (ze <-> si))
dedth4h.4	|- (D = if(th, D, G) -> (si <-> rh))
dedth4h.5	|- rh

Assertion

Ref	Expression
dedth4h	|- (((ph /\ ps) /\ (ch /\ th)) -> ta )

Proof of Theorem dedth4h

Step	Hyp	Ref	Expression
1		dedth4h.1	. . . 4 |- (A = if(ph, A, R) -> (ta <-> et))
2	1	imbi2d 464	. . 3 |- (A = if(ph, A, R) -> (((ch /\ th) -> ta ) <-> ((ch /\ th) -> et)))
3		dedth4h.2	. . . 4 |- (B = if(ps, B, S) -> (et <-> ze))
4	3	imbi2d 464	. . 3 |- (B = if(ps, B, S) -> (((ch /\ th) -> et) <-> ((ch /\ th) -> ze)))
5		dedth4h.3	. . . 4 |- (C = if(ch, C, F) -> (ze <-> si))
6		dedth4h.4	. . . 4 |- (D = if(th, D, G) -> (si <-> rh))
7		dedth4h.5	. . . 4 |- rh
8	5, 6, 7	dedth2h 1787	. . 3 |- ((ch /\ th) -> ze)
9	2, 4, 8	dedth2h 1787	. 2 |- ((ph /\ ps) -> ((ch /\ th) -> ta ))
10	9	imp 277	1 |- (((ph /\ ps) /\ (ch /\ th)) -> ta )

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196   = wceq 1091  ifcif 1776

This theorem is referenced by:  lt2addt 4361  crut 4531  nn0opth2t 4726  abs3lemt 4865  hvsubsub4t 5032  norm3lemt 5097  projlem20 5212

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-if 1777

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