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Mathbox for Andrew Salmon |
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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 2493 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝑥 = 𝑧) → [𝑦 / 𝑥]𝑥 = 𝑧) | |
2 | equsb3 2106 | . 2 ⊢ ([𝑦 / 𝑥]𝑥 = 𝑧 ↔ 𝑦 = 𝑧) | |
3 | 1, 2 | sylib 221 | 1 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝑥 = 𝑧) → 𝑦 = 𝑧) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1536 [wsb 2069 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2142 ax-12 2175 ax-13 2379 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-ex 1782 df-nf 1786 df-sb 2070 |
This theorem is referenced by: sbeqal1i 41103 |
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