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Mirrors > Home > MPE Home > Th. List > Mathboxes > sbeqal2i | Structured version Visualization version GIF version |
Description: If 𝑥 = 𝑦 implies 𝑥 = 𝑧, then we can infer 𝑧 = 𝑦. (Contributed by Andrew Salmon, 3-Jun-2011.) |
Ref | Expression |
---|---|
sbeqal1i.1 | ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑧) |
Ref | Expression |
---|---|
sbeqal2i | ⊢ 𝑧 = 𝑦 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbeqal1i.1 | . . 3 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑧) | |
2 | 1 | sbeqal1i 41555 | . 2 ⊢ 𝑦 = 𝑧 |
3 | 2 | eqcomi 2747 | 1 ⊢ 𝑧 = 𝑦 |
Colors of variables: wff setvar class |
Syntax hints: |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-9 2124 ax-10 2145 ax-12 2179 ax-13 2372 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-ex 1787 df-nf 1791 df-sb 2075 df-cleq 2730 |
This theorem is referenced by: (None) |
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