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Theorem dedt 1024
Description: The weak deduction theorem. For more information, see the Weak Deduction Theorem page mmdeduction.html. (Contributed by NM, 26-Jun-2002.) Revised to use the conditional operator. (Revised by BJ, 30-Sep-2019.)
Hypotheses
Ref Expression
dedt.1 ((if-(𝜒, 𝜑, 𝜓) ↔ 𝜑) → (𝜃𝜏))
dedt.2 𝜏
Assertion
Ref Expression
dedt (𝜒𝜃)

Proof of Theorem dedt
StepHypRef Expression
1 ifptru 1016 . 2 (𝜒 → (if-(𝜒, 𝜑, 𝜓) ↔ 𝜑))
2 dedt.2 . . 3 𝜏
3 dedt.1 . . 3 ((if-(𝜒, 𝜑, 𝜓) ↔ 𝜑) → (𝜃𝜏))
42, 3mpbiri 246 . 2 ((if-(𝜒, 𝜑, 𝜓) ↔ 𝜑) → 𝜃)
51, 4syl 17 1 (𝜒𝜃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  if-wif 1005
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-or 383  df-an 384  df-ifp 1006
This theorem is referenced by:  con3ALT  1025
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