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Theorem dedth4h 4520
Description: Weak deduction theorem eliminating four hypotheses. See comments in dedth2h 4518. (Contributed by NM, 16-May-1999.)
Hypotheses
Ref Expression
dedth4h.1 (𝐴 = if(𝜑, 𝐴, 𝑅) → (𝜏𝜂))
dedth4h.2 (𝐵 = if(𝜓, 𝐵, 𝑆) → (𝜂𝜁))
dedth4h.3 (𝐶 = if(𝜒, 𝐶, 𝐹) → (𝜁𝜎))
dedth4h.4 (𝐷 = if(𝜃, 𝐷, 𝐺) → (𝜎𝜌))
dedth4h.5 𝜌
Assertion
Ref Expression
dedth4h (((𝜑𝜓) ∧ (𝜒𝜃)) → 𝜏)

Proof of Theorem dedth4h
StepHypRef Expression
1 dedth4h.1 . . . 4 (𝐴 = if(𝜑, 𝐴, 𝑅) → (𝜏𝜂))
21imbi2d 341 . . 3 (𝐴 = if(𝜑, 𝐴, 𝑅) → (((𝜒𝜃) → 𝜏) ↔ ((𝜒𝜃) → 𝜂)))
3 dedth4h.2 . . . 4 (𝐵 = if(𝜓, 𝐵, 𝑆) → (𝜂𝜁))
43imbi2d 341 . . 3 (𝐵 = if(𝜓, 𝐵, 𝑆) → (((𝜒𝜃) → 𝜂) ↔ ((𝜒𝜃) → 𝜁)))
5 dedth4h.3 . . . 4 (𝐶 = if(𝜒, 𝐶, 𝐹) → (𝜁𝜎))
6 dedth4h.4 . . . 4 (𝐷 = if(𝜃, 𝐷, 𝐺) → (𝜎𝜌))
7 dedth4h.5 . . . 4 𝜌
85, 6, 7dedth2h 4518 . . 3 ((𝜒𝜃) → 𝜁)
92, 4, 8dedth2h 4518 . 2 ((𝜑𝜓) → ((𝜒𝜃) → 𝜏))
109imp 407 1 (((𝜑𝜓) ∧ (𝜒𝜃)) → 𝜏)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539  ifcif 4459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-if 4460
This theorem is referenced by:  dedth4v  4523  fprg  7027  omopth  8492  nn0opth2  13986  ax5seglem8  27304  hvsubsub4  29422  norm3lemt  29514  eigorth  30200
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