New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  elimh GIF version

Theorem elimh 922
 Description: Hypothesis builder for weak deduction theorem. For more information, see the Deduction Theorem link on the Metamath Proof Explorer home page. (Contributed by NM, 26-Jun-2002.)
Hypotheses
Ref Expression
elimh.1 ((φ ↔ ((φ χ) (ψ ¬ χ))) → (χτ))
elimh.2 ((ψ ↔ ((φ χ) (ψ ¬ χ))) → (θτ))
elimh.3 θ
Assertion
Ref Expression
elimh τ

Proof of Theorem elimh
StepHypRef Expression
1 dedlema 920 . . . 4 (χ → (φ ↔ ((φ χ) (ψ ¬ χ))))
2 elimh.1 . . . 4 ((φ ↔ ((φ χ) (ψ ¬ χ))) → (χτ))
31, 2syl 15 . . 3 (χ → (χτ))
43ibi 232 . 2 (χτ)
5 elimh.3 . . 3 θ
6 dedlemb 921 . . . 4 χ → (ψ ↔ ((φ χ) (ψ ¬ χ))))
7 elimh.2 . . . 4 ((ψ ↔ ((φ χ) (ψ ¬ χ))) → (θτ))
86, 7syl 15 . . 3 χ → (θτ))
95, 8mpbii 202 . 2 χτ)
104, 9pm2.61i 156 1 τ
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360 This theorem is referenced by:  con3th  924
 Copyright terms: Public domain W3C validator