MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dedtOLD Structured version   Visualization version   GIF version

Theorem dedtOLD 1075
Description: Obsolete version of dedt 1074 as of 27-Apr-2023. The weak deduction theorem. For more information, see the Weak Deduction Theorem page mmdeduction.html 1074. (Contributed by NM, 26-Jun-2002.) Revised to use the conditional operator. (Revised by BJ, 30-Sep-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
dedtOLD.1 ((if-(𝜒, 𝜑, 𝜓) ↔ 𝜑) → (𝜃𝜏))
dedtOLD.2 𝜏
Assertion
Ref Expression
dedtOLD (𝜒𝜃)

Proof of Theorem dedtOLD
StepHypRef Expression
1 ifptru 1065 . 2 (𝜒 → (if-(𝜒, 𝜑, 𝜓) ↔ 𝜑))
2 dedtOLD.2 . . 3 𝜏
3 dedtOLD.1 . . 3 ((if-(𝜒, 𝜑, 𝜓) ↔ 𝜑) → (𝜃𝜏))
42, 3mpbiri 259 . 2 ((if-(𝜒, 𝜑, 𝜓) ↔ 𝜑) → 𝜃)
51, 4syl 17 1 (𝜒𝜃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  if-wif 1054
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 208  df-an 397  df-or 842  df-ifp 1055
This theorem is referenced by:  con3ALTOLD  1077
  Copyright terms: Public domain W3C validator