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Mirrors > Home > MPE Home > Th. List > equvelv | Structured version Visualization version GIF version |
Description: A biconditional form of equvel 2458 with disjoint variable conditions and proved from Tarski's FOL axiom schemes. (Contributed by Andrew Salmon, 2-Jun-2011.) Reduce axiom usage. (Revised by Wolf Lammen, 10-Apr-2021.) (Proof shortened by Wolf Lammen, 12-Jul-2022.) |
Ref | Expression |
---|---|
equvelv | ⊢ (∀𝑧(𝑧 = 𝑥 → 𝑧 = 𝑦) ↔ 𝑥 = 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equequ1 2032 | . 2 ⊢ (𝑧 = 𝑥 → (𝑧 = 𝑦 ↔ 𝑥 = 𝑦)) | |
2 | 1 | equsalvw 2011 | 1 ⊢ (∀𝑧(𝑧 = 𝑥 → 𝑧 = 𝑦) ↔ 𝑥 = 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1787 |
This theorem is referenced by: eu6lem 2575 |
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