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Theorem dedth3h 3705
 Description: Weak deduction theorem eliminating three hypotheses. See comments in dedth2h 3704. (Contributed by NM, 15-May-1999.)
Hypotheses
Ref Expression
dedth3h.1 (A = if(φ, A, D) → (θτ))
dedth3h.2 (B = if(ψ, B, R) → (τη))
dedth3h.3 (C = if(χ, C, S) → (ηζ))
dedth3h.4 ζ
Assertion
Ref Expression
dedth3h ((φ ψ χ) → θ)

Proof of Theorem dedth3h
StepHypRef Expression
1 dedth3h.1 . . . 4 (A = if(φ, A, D) → (θτ))
21imbi2d 307 . . 3 (A = if(φ, A, D) → (((ψ χ) → θ) ↔ ((ψ χ) → τ)))
3 dedth3h.2 . . . 4 (B = if(ψ, B, R) → (τη))
4 dedth3h.3 . . . 4 (C = if(χ, C, S) → (ηζ))
5 dedth3h.4 . . . 4 ζ
63, 4, 5dedth2h 3704 . . 3 ((ψ χ) → τ)
72, 6dedth 3703 . 2 (φ → ((ψ χ) → θ))
873impib 1149 1 ((φ ψ χ) → θ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934   = wceq 1642   ifcif 3662 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-if 3663 This theorem is referenced by:  dedth3v  3708
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