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| Mirrors > Home > MPE Home > Th. List > equcomd | Structured version Visualization version GIF version | ||
| Description: Deduction form of equcom 2017, symmetry of equality. For the versions for classes, see eqcom 2744 and eqcomd 2743. (Contributed by BJ, 6-Oct-2019.) |
| Ref | Expression |
|---|---|
| equcomd.1 | ⊢ (𝜑 → 𝑥 = 𝑦) |
| Ref | Expression |
|---|---|
| equcomd | ⊢ (𝜑 → 𝑦 = 𝑥) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | equcomd.1 | . 2 ⊢ (𝜑 → 𝑥 = 𝑦) | |
| 2 | equcom 2017 | . 2 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥) | |
| 3 | 1, 2 | sylib 218 | 1 ⊢ (𝜑 → 𝑦 = 𝑥) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 |
| This theorem is referenced by: sndisj 5135 fsumcom2 15810 fprodcom2 16020 catideu 17718 pospo 18390 dprdfcntz 20035 ordtt1 23387 eengtrkg 29001 cusgrfilem2 29474 frgr2wwlk1 30348 ssmxidl 33502 gonar 35400 bj-nfcsym 36900 exidu1 37863 rngoideu 37910 2reu8i 47125 ichnreuop 47459 sprsymrelf1lem 47478 oppcthinendcALT 49090 |
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