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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 2020 | . 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 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 |
| This theorem is referenced by: equeuclr 2022 equequ2 2025 ax12v2 2179 2ax6elem 2475 axprlem3OLD 5428 wl-spae 37522 ax12eq 38942 sn-axprlem3 42256 ax6e2eq 44577 ax6e2eqVD 44927 |
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