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Mirrors > Home > ILE Home > Th. List > equcom | GIF version |
Description: Commutative law for equality. (Contributed by NM, 20-Aug-1993.) |
Ref | Expression |
---|---|
equcom | ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equcomi 1681 | . 2 ⊢ (𝑥 = 𝑦 → 𝑦 = 𝑥) | |
2 | equcomi 1681 | . 2 ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦) | |
3 | 1, 2 | impbii 125 | 1 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-gen 1426 ax-ie2 1471 ax-8 1483 ax-17 1507 ax-i9 1511 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: equcomd 1684 sbal1yz 1977 dveeq1 1995 eu1 2025 reu7 2883 reu8 2884 dfdif3 3191 iunid 3876 copsexg 4174 opelopabsbALT 4189 dtruex 4482 opeliunxp 4602 relop 4697 dmi 4762 opabresid 4880 intirr 4933 cnvi 4951 coi1 5062 brprcneu 5422 f1oiso 5735 qsid 6502 mapsnen 6713 suplocsrlem 7640 summodc 11184 bezoutlemle 11732 cnmptid 12489 |
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