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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege55lem1a | Structured version Visualization version GIF version |
Description: Necessary deduction regarding substitution of value in equality. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege55lem1a | ⊢ ((𝜏 → if-(𝜓, 𝜑, ¬ 𝜑)) → (𝜏 → (𝜓 ↔ 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege54cor0a 41452 | . . 3 ⊢ ((𝜓 ↔ 𝜑) ↔ if-(𝜓, 𝜑, ¬ 𝜑)) | |
2 | 1 | biimpri 227 | . 2 ⊢ (if-(𝜓, 𝜑, ¬ 𝜑) → (𝜓 ↔ 𝜑)) |
3 | 2 | imim2i 16 | 1 ⊢ ((𝜏 → if-(𝜓, 𝜑, ¬ 𝜑)) → (𝜏 → (𝜓 ↔ 𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 if-wif 1060 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-frege28 41419 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-ifp 1061 |
This theorem is referenced by: frege55cor1a 41458 |
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