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Theorem elimh 1072
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 1065 . . . 4 (𝜒 → (if-(𝜒, 𝜑, 𝜓) ↔ 𝜑))
2 elimh.1 . . . 4 ((if-(𝜒, 𝜑, 𝜓) ↔ 𝜑) → (𝜏𝜒))
31, 2syl 17 . . 3 (𝜒 → (𝜏𝜒))
43ibir 269 . 2 (𝜒𝜏)
5 elimh.3 . . 3 𝜃
6 ifpfal 1066 . . . 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 259 . 2 𝜒𝜏)
104, 9pm2.61i 183 1 𝜏
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  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:  con3ALT  1076
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