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Theorem dedth4h 4092
 Description: Weak deduction theorem eliminating four hypotheses. See comments in dedth2h 4090. (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 329 . . 3 (𝐴 = if(𝜑, 𝐴, 𝑅) → (((𝜒𝜃) → 𝜏) ↔ ((𝜒𝜃) → 𝜂)))
3 dedth4h.2 . . . 4 (𝐵 = if(𝜓, 𝐵, 𝑆) → (𝜂𝜁))
43imbi2d 329 . . 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 4090 . . 3 ((𝜒𝜃) → 𝜁)
92, 4, 8dedth2h 4090 . 2 ((𝜑𝜓) → ((𝜒𝜃) → 𝜏))
109imp 444 1 (((𝜑𝜓) ∧ (𝜒𝜃)) → 𝜏)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ifcif 4036 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-if 4037 This theorem is referenced by:  dedth4v  4095  fprg  6327  omopth  7625  nn0opth2  12921  ax5seglem8  25616  hvsubsub4  27301  norm3lemt  27393  eigorth  28081
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