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Mirrors > Home > NFE Home > Th. List > equequ2 | GIF version |
Description: An equivalence law for equality. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Aug-2017.) |
Ref | Expression |
---|---|
equequ2 | ⊢ (x = y → (z = x ↔ z = y)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equequ1 1684 | . 2 ⊢ (x = y → (x = z ↔ y = z)) | |
2 | equcom 1680 | . 2 ⊢ (x = z ↔ z = x) | |
3 | equcom 1680 | . 2 ⊢ (y = z ↔ z = y) | |
4 | 1, 2, 3 | 3bitr3g 278 | 1 ⊢ (x = y → (z = x ↔ z = y)) |
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: ax10lem2 1937 dveeq2 1940 ax10lem4 1941 ax9 1949 ax11v2 1992 ax11vALT 2097 dveeq2-o 2184 dveeq2-o16 2185 ax10-16 2190 ax11eq 2193 ax11inda 2200 ax11v2-o 2201 eujust 2206 eujustALT 2207 euf 2210 mo 2226 2mo 2282 2eu6 2289 iotaval 4350 nndisjeq 4429 dfid3 4768 |
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