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Mirrors > Home > MPE Home > Th. List > bicom | Structured version Visualization version GIF version |
Description: Commutative law for the biconditional. Theorem *4.21 of [WhiteheadRussell] p. 117. (Contributed by NM, 11-May-1993.) |
Ref | Expression |
---|---|
bicom | ⊢ ((𝜑 ↔ 𝜓) ↔ (𝜓 ↔ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bicom1 210 | . 2 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 ↔ 𝜑)) | |
2 | bicom1 210 | . 2 ⊢ ((𝜓 ↔ 𝜑) → (𝜑 ↔ 𝜓)) | |
3 | 1, 2 | impbii 198 | 1 ⊢ ((𝜑 ↔ 𝜓) ↔ (𝜓 ↔ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 |
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 196 |
This theorem is referenced by: bicomd 212 bibi1i 327 bibi1d 332 con2bi 342 ibibr 357 bibif 360 nbbn 372 pm5.17 928 biluk 970 bigolden 972 xorcom 1459 trubifal 1513 exists1 2549 abeq1 2720 ssequn1 3745 axpow3 4772 isocnv 6480 qextlt 11908 qextle 11909 rpnnen2lem12 14793 odd2np1 14903 sumodd 14949 nrmmetd 22189 lgsqrmodndvds 24878 cusgrauvtxb 26024 cvmlift2lem12 30550 nn0prpw 31488 bj-abeq1 31962 tsbi4 33113 bicomdd 33153 ifpbicor 36839 rp-fakeoranass 36878 or3or 37339 3impexpbicom 37706 3impexpbicomVD 38114 nabctnabc 39747 |
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