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Theorem dedth4h 4279
Description: Weak deduction theorem eliminating four hypotheses. See comments in dedth2h 4277. (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 4277 . . 3 ((𝜒𝜃) → 𝜁)
92, 4, 8dedth2h 4277 . 2 ((𝜑𝜓) → ((𝜒𝜃) → 𝜏))
109imp 393 1 (((𝜑𝜓) ∧ (𝜒𝜃)) → 𝜏)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1630  ifcif 4223
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-if 4224
This theorem is referenced by:  dedth4v  4282  fprg  6564  omopth  7891  nn0opth2  13262  ax5seglem8  26036  hvsubsub4  28251  norm3lemt  28343  eigorth  29031
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