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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 1935 | . 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 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 |
This theorem is referenced by: equeuclr 1937 equequ2 1940 equtr2OLD 1943 equvinv 1946 equvinivOLD 1948 equvinvOLD 1949 equvelv 1950 ax12v2 2036 ax12vOLD 2037 2ax6elem 2437 wl-spae 32485 ax12eq 33244 sbeqalbi 37623 ax6e2eq 37794 ax6e2eqVD 38165 |
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