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Mirrors > Home > MPE Home > Th. List > equtrr | Structured version Visualization version GIF version |
Description: A transitive law for equality. Lemma L17 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 23-Aug-1993.) |
Ref | Expression |
---|---|
equtrr | ⊢ (𝑥 = 𝑦 → (𝑧 = 𝑥 → 𝑧 = 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equtr 2018 | . 2 ⊢ (𝑧 = 𝑥 → (𝑥 = 𝑦 → 𝑧 = 𝑦)) | |
2 | 1 | com12 32 | 1 ⊢ (𝑥 = 𝑦 → (𝑧 = 𝑥 → 𝑧 = 𝑦)) |
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 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 |
This theorem is referenced by: equeuclr 2020 equequ2 2023 ax12v2 2177 2ax6elem 2473 axprlem3OLD 5434 wl-spae 37502 ax12eq 38923 sn-axprlem3 42235 ax6e2eq 44555 ax6e2eqVD 44905 |
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