Theorem dedth4h 2387

Description: Weak deduction theorem eliminating four hypotheses. See comments in dedth2h 2385.

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 611	. . 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 611	. . 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 2385	. . 3 |- ((ch /\ th) -> ze)
9	2, 4, 8	dedth2h 2385	. 2 |- ((ph /\ ps) -> ((ch /\ th) -> ta))
10	9	imp 350	1 |- (((ph /\ ps) /\ (ch /\ th)) -> ta)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 955  ifcif 2359

This theorem is referenced by:  lt2addt 5631  le2addt 5632  nn0opth2t 6619  crut 6689  abs3lemt 6873  hvsubsub4t 8911  norm3lemt 9003  projlem20 9193  eigortht 9755

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-if 2360

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