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Mirrors > Home > ILE Home > Th. List > eqtr | GIF version |
Description: Transitive law for class equality. Proposition 4.7(3) of [TakeutiZaring] p. 13. (Contributed by NM, 25-Jan-2004.) |
Ref | Expression |
---|---|
eqtr | ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐶) → 𝐴 = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2147 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐶)) | |
2 | 1 | biimpar 295 | 1 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐶) → 𝐴 = 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1332 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1424 ax-gen 1426 ax-4 1488 ax-17 1507 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-cleq 2133 |
This theorem is referenced by: eqtr2 2159 eqtr3 2160 sylan9eq 2193 eqvinc 2812 eqvincg 2813 uneqdifeqim 3453 preqsn 3710 dtruex 4482 relresfld 5076 relcoi1 5078 eqer 6469 xpider 6508 addlsub 8156 bj-findis 13348 |
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