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Mirrors > Home > MPE Home > Th. List > equcomiv | Structured version Visualization version GIF version |
Description: Weaker form of equcomi 1931 with a dv condition on 𝑥, 𝑦. This is an intermediate step and equcomi 1931 is fully recovered later. (Contributed by BJ, 7-Dec-2020.) |
Ref | Expression |
---|---|
equcomiv | ⊢ (𝑥 = 𝑦 → 𝑦 = 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equid 1926 | . 2 ⊢ 𝑥 = 𝑥 | |
2 | ax7v2 1925 | . 2 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑥 → 𝑦 = 𝑥)) | |
3 | 1, 2 | mpi 20 | 1 ⊢ (𝑥 = 𝑦 → 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 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 |
This theorem depends on definitions: df-bi 196 df-ex 1696 |
This theorem is referenced by: ax6evr 1929 |
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