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Mirrors > Home > MPE Home > Th. List > equvinv | Structured version Visualization version GIF version |
Description: A variable introduction law for equality. Lemma 15 of [Monk2] p. 109. (Contributed by NM, 9-Jan-1993.) Remove dependencies on ax-10 2142, ax-13 2379. (Revised by Wolf Lammen, 10-Jun-2019.) Move the quantified variable (𝑧) to the left of the equality signs. (Revised by Wolf Lammen, 11-Apr-2021.) (Proof shortened by Wolf Lammen, 12-Jul-2022.) |
Ref | Expression |
---|---|
equvinv | ⊢ (𝑥 = 𝑦 ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 = 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equequ1 2032 | . . 3 ⊢ (𝑧 = 𝑥 → (𝑧 = 𝑦 ↔ 𝑥 = 𝑦)) | |
2 | 1 | equsexvw 2011 | . 2 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ 𝑧 = 𝑦) ↔ 𝑥 = 𝑦) |
3 | 2 | bicomi 227 | 1 ⊢ (𝑥 = 𝑦 ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 = 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∃wex 1781 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 |
This theorem is referenced by: ax8 2117 ax9 2125 ax13 2382 cossid 35880 |
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