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Mirrors > Home > MPE Home > Th. List > Mathboxes > ifpbicor | Structured version Visualization version GIF version |
Description: Corollary of commutation of biimplication. (Contributed by RP, 25-Apr-2020.) |
Ref | Expression |
---|---|
ifpbicor | ⊢ (if-(𝜑, 𝜓, ¬ 𝜓) ↔ if-(𝜓, 𝜑, ¬ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bicom 224 | . 2 ⊢ ((𝜑 ↔ 𝜓) ↔ (𝜓 ↔ 𝜑)) | |
2 | ifpdfbi 39859 | . 2 ⊢ ((𝜑 ↔ 𝜓) ↔ if-(𝜑, 𝜓, ¬ 𝜓)) | |
3 | ifpdfbi 39859 | . 2 ⊢ ((𝜓 ↔ 𝜑) ↔ if-(𝜓, 𝜑, ¬ 𝜑)) | |
4 | 1, 2, 3 | 3bitr3i 303 | 1 ⊢ (if-(𝜑, 𝜓, ¬ 𝜓) ↔ if-(𝜓, 𝜑, ¬ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ 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 df-tru 1540 |
This theorem is referenced by: ifpxorcor 39862 |
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