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Mirrors > Home > MPE Home > Th. List > Mathboxes > sbeqal1 | Structured version Visualization version GIF version |
Description: If 𝑥 = 𝑦 always implies 𝑥 = 𝑧, then 𝑦 = 𝑧. (Contributed by Andrew Salmon, 2-Jun-2011.) |
Ref | Expression |
---|---|
sbeqal1 | ⊢ (∀𝑥(𝑥 = 𝑦 → 𝑥 = 𝑧) → 𝑦 = 𝑧) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb2 2497 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝑥 = 𝑧) → [𝑦 / 𝑥]𝑥 = 𝑧) | |
2 | equsb3 2100 | . 2 ⊢ ([𝑦 / 𝑥]𝑥 = 𝑧 ↔ 𝑦 = 𝑧) | |
3 | 1, 2 | sylib 219 | 1 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝑥 = 𝑧) → 𝑦 = 𝑧) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1526 [wsb 2060 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-10 2136 ax-12 2167 ax-13 2381 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-ex 1772 df-nf 1776 df-sb 2061 |
This theorem is referenced by: sbeqal1i 40608 |
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