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Theorem elimh 1080
 Description: Hypothesis builder for 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.) Commute consequent. (Revised by Steven Nguyen, 27-Apr-2023.)
Hypotheses
Ref Expression
elimh.1 ((if-(𝜒, 𝜑, 𝜓) ↔ 𝜑) → (𝜏𝜒))
elimh.2 ((if-(𝜒, 𝜑, 𝜓) ↔ 𝜓) → (𝜏𝜃))
elimh.3 𝜃
Assertion
Ref Expression
elimh 𝜏

Proof of Theorem elimh
StepHypRef Expression
1 ifptru 1071 . . . 4 (𝜒 → (if-(𝜒, 𝜑, 𝜓) ↔ 𝜑))
2 elimh.1 . . . 4 ((if-(𝜒, 𝜑, 𝜓) ↔ 𝜑) → (𝜏𝜒))
31, 2syl 17 . . 3 (𝜒 → (𝜏𝜒))
43ibir 271 . 2 (𝜒𝜏)
5 elimh.3 . . 3 𝜃
6 ifpfal 1072 . . . 4 𝜒 → (if-(𝜒, 𝜑, 𝜓) ↔ 𝜓))
7 elimh.2 . . . 4 ((if-(𝜒, 𝜑, 𝜓) ↔ 𝜓) → (𝜏𝜃))
86, 7syl 17 . . 3 𝜒 → (𝜏𝜃))
95, 8mpbiri 261 . 2 𝜒𝜏)
104, 9pm2.61i 185 1 𝜏
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209  if-wif 1058 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 210  df-an 400  df-or 845  df-ifp 1059 This theorem is referenced by:  con3ALT  1082
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