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Theorem dedlem0b 991
Description: Lemma for an alternate version of weak deduction theorem. (Contributed by NM, 2-Apr-1994.)
Assertion
Ref Expression
dedlem0b 𝜑 → (𝜓 ↔ ((𝜓𝜑) → (𝜒𝜑))))

Proof of Theorem dedlem0b
StepHypRef Expression
1 pm2.21 118 . . . 4 𝜑 → (𝜑 → (𝜒𝜑)))
21imim2d 54 . . 3 𝜑 → ((𝜓𝜑) → (𝜓 → (𝜒𝜑))))
32com23 83 . 2 𝜑 → (𝜓 → ((𝜓𝜑) → (𝜒𝜑))))
4 pm2.21 118 . . . . 5 𝜓 → (𝜓𝜑))
5 simpr 475 . . . . 5 ((𝜒𝜑) → 𝜑)
64, 5imim12i 59 . . . 4 (((𝜓𝜑) → (𝜒𝜑)) → (¬ 𝜓𝜑))
76con1d 137 . . 3 (((𝜓𝜑) → (𝜒𝜑)) → (¬ 𝜑𝜓))
87com12 32 . 2 𝜑 → (((𝜓𝜑) → (𝜒𝜑)) → 𝜓))
93, 8impbid 200 1 𝜑 → (𝜓 ↔ ((𝜓𝜑) → (𝜒𝜑))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382
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 195  df-an 384
This theorem is referenced by: (None)
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