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Mirrors > Home > MPE Home > Th. List > elimhOLD | Structured version Visualization version GIF version |
Description: Obsolete version of elimh 1076 as of 27-Apr-2023. Hypothesis builder for the weak deduction theorem. For more information, see the Weak Deduction Theorem page mmdeduction.html 1076. (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.) |
Ref | Expression |
---|---|
elimhOLD.1 | ⊢ ((if-(𝜒, 𝜑, 𝜓) ↔ 𝜑) → (𝜒 ↔ 𝜏)) |
elimhOLD.2 | ⊢ ((if-(𝜒, 𝜑, 𝜓) ↔ 𝜓) → (𝜃 ↔ 𝜏)) |
elimhOLD.3 | ⊢ 𝜃 |
Ref | Expression |
---|---|
elimhOLD | ⊢ 𝜏 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ifptru 1068 | . . . 4 ⊢ (𝜒 → (if-(𝜒, 𝜑, 𝜓) ↔ 𝜑)) | |
2 | elimhOLD.1 | . . . 4 ⊢ ((if-(𝜒, 𝜑, 𝜓) ↔ 𝜑) → (𝜒 ↔ 𝜏)) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜒 → (𝜒 ↔ 𝜏)) |
4 | 3 | ibi 269 | . 2 ⊢ (𝜒 → 𝜏) |
5 | elimhOLD.3 | . . 3 ⊢ 𝜃 | |
6 | ifpfal 1069 | . . . 4 ⊢ (¬ 𝜒 → (if-(𝜒, 𝜑, 𝜓) ↔ 𝜓)) | |
7 | elimhOLD.2 | . . . 4 ⊢ ((if-(𝜒, 𝜑, 𝜓) ↔ 𝜓) → (𝜃 ↔ 𝜏)) | |
8 | 6, 7 | syl 17 | . . 3 ⊢ (¬ 𝜒 → (𝜃 ↔ 𝜏)) |
9 | 5, 8 | mpbii 235 | . 2 ⊢ (¬ 𝜒 → 𝜏) |
10 | 4, 9 | pm2.61i 184 | 1 ⊢ 𝜏 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 if-wif 1057 |
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 209 df-an 399 df-or 844 df-ifp 1058 |
This theorem is referenced by: con3ALTOLD 1081 |
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