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Mirrors > Home > MPE Home > Th. List > dedt | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
dedt.1 | ⊢ ((if-(𝜒, 𝜑, 𝜓) ↔ 𝜑) → (𝜃 ↔ 𝜏)) |
dedt.2 | ⊢ 𝜏 |
Ref | Expression |
---|---|
dedt | ⊢ (𝜒 → 𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ifptru 1023 | . 2 ⊢ (𝜒 → (if-(𝜒, 𝜑, 𝜓) ↔ 𝜑)) | |
2 | dedt.2 | . . 3 ⊢ 𝜏 | |
3 | dedt.1 | . . 3 ⊢ ((if-(𝜒, 𝜑, 𝜓) ↔ 𝜑) → (𝜃 ↔ 𝜏)) | |
4 | 2, 3 | mpbiri 248 | . 2 ⊢ ((if-(𝜒, 𝜑, 𝜓) ↔ 𝜑) → 𝜃) |
5 | 1, 4 | syl 17 | 1 ⊢ (𝜒 → 𝜃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 if-wif 1012 |
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 197 df-or 385 df-an 386 df-ifp 1013 |
This theorem is referenced by: con3ALT 1032 |
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