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Mirrors > Home > NFE Home > Th. List > equequ1 | GIF version |
Description: An equivalence law for equality. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 10-Dec-2017.) |
Ref | Expression |
---|---|
equequ1 | ⊢ (x = y → (x = z ↔ y = z)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-8 1675 | . 2 ⊢ (x = y → (x = z → y = z)) | |
2 | equtr 1682 | . 2 ⊢ (x = y → (y = z → x = z)) | |
3 | 1, 2 | impbid 183 | 1 ⊢ (x = y → (x = z ↔ y = z)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 |
This theorem depends on definitions: df-bi 177 df-ex 1542 |
This theorem is referenced by: equequ2 1686 ax12olem6 1932 ax10lem2 1937 ax10lem4 1941 equveli 1988 dveeq1 2018 drsb1 2022 equsb3lem 2101 dveeq1-o 2187 dveeq1-o16 2188 ax10-16 2190 ax11eq 2193 2mo 2282 2eu6 2289 euequ1 2292 axext3 2336 cbviota 4344 |
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