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Mirrors > Home > MPE Home > Th. List > equcomd | Structured version Visualization version GIF version |
Description: Deduction form of equcom 2022, symmetry of equality. For the versions for classes, see eqcom 2740 and eqcomd 2739. (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 2022 | . 2 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥) | |
3 | 1, 2 | sylib 217 | 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 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 |
This theorem is referenced by: sndisj 5140 fsumcom2 15720 fprodcom2 15928 catideu 17619 pospo 18298 dprdfcntz 19885 ordtt1 22883 eengtrkg 28244 cusgrfilem2 28713 frgr2wwlk1 29582 ssmxidl 32590 gonar 34386 bj-nfcsym 35779 exidu1 36724 rngoideu 36771 2reu8i 45821 ichnreuop 46140 sprsymrelf1lem 46159 |
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