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Mirrors > Home > MPE Home > Th. List > bicom1 | Structured version Visualization version GIF version |
Description: Commutative law for the biconditional. (Contributed by Wolf Lammen, 10-Nov-2012.) |
Ref | Expression |
---|---|
bicom1 | ⊢ ((𝜑 ↔ 𝜓) → (𝜓 ↔ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | biimpr 212 | . 2 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑)) | |
2 | biimp 207 | . 2 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓)) | |
3 | 1, 2 | impbid 204 | 1 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 ↔ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 |
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 199 |
This theorem is referenced by: bicom 214 bicomi 216 con3ALTOLD 1069 nanass 1580 nanassOLD 1581 frege55aid 39115 frege55lem2a 39117 bisaiaisb 42007 confun4 42036 confun5 42037 |
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