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Mirrors > Home > ILE Home > Th. List > eqtr2 | GIF version |
Description: A transitive law for class equality. (Contributed by NM, 20-May-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Ref | Expression |
---|---|
eqtr2 | ⊢ ((A = B ∧ A = 𝐶) → B = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqcom 2039 | . 2 ⊢ (A = B ↔ B = A) | |
2 | eqtr 2054 | . 2 ⊢ ((B = A ∧ A = 𝐶) → B = 𝐶) | |
3 | 1, 2 | sylanb 268 | 1 ⊢ ((A = B ∧ A = 𝐶) → B = 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1242 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1333 ax-gen 1335 ax-4 1397 ax-17 1416 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-cleq 2030 |
This theorem is referenced by: eqvinc 2661 eqvincg 2662 moop2 3979 reusv3i 4157 relop 4429 fliftfun 5379 th3qlem1 6144 enq0ref 6416 enq0tr 6417 genpdisj 6506 |
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