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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 219 | . 2 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑)) | |
2 | biimp 214 | . 2 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓)) | |
3 | 1, 2 | impbid 211 | 1 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 ↔ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 |
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 206 |
This theorem is referenced by: bicom 221 bicomi 223 nanass 1505 rexsng 4611 nummin 33049 bicomdALT 40150 frege55aid 41432 frege55lem2a 41434 bisaiaisb 44364 confun4 44393 confun5 44394 ichnfim 44872 |
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