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Mirrors > Home > MPE Home > Th. List > eqtr2OLD | Structured version Visualization version GIF version |
Description: Obsolete version of eqtr2 as of 24-Oct-2024. (Contributed by NM, 20-May-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
eqtr2OLD | ⊢ ((𝐴 = 𝐵 ∧ 𝐴 = 𝐶) → 𝐵 = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqcom 2740 | . 2 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴) | |
2 | eqtr 2756 | . 2 ⊢ ((𝐵 = 𝐴 ∧ 𝐴 = 𝐶) → 𝐵 = 𝐶) | |
3 | 1, 2 | sylanb 582 | 1 ⊢ ((𝐴 = 𝐵 ∧ 𝐴 = 𝐶) → 𝐵 = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 df-cleq 2725 |
This theorem is referenced by: (None) |
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