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Mirrors > Home > MPE Home > Th. List > equvinvOLD | Structured version Visualization version GIF version |
Description: Obsolete version of equvinv 1959 as of 11-Apr-2021. (Contributed by NM, 9-Jan-1993.) Remove dependencies on ax-10 2019, ax-13 2246. (Revised by Wolf Lammen, 10-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
equvinvOLD | ⊢ (𝑥 = 𝑦 ↔ ∃𝑧(𝑥 = 𝑧 ∧ 𝑦 = 𝑧)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equviniva 1960 | . 2 ⊢ (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧 ∧ 𝑦 = 𝑧)) | |
2 | equtrr 1949 | . . . . 5 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑧 → 𝑥 = 𝑦)) | |
3 | 2 | equcoms 1947 | . . . 4 ⊢ (𝑦 = 𝑧 → (𝑥 = 𝑧 → 𝑥 = 𝑦)) |
4 | 3 | impcom 446 | . . 3 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑧) → 𝑥 = 𝑦) |
5 | 4 | exlimiv 1858 | . 2 ⊢ (∃𝑧(𝑥 = 𝑧 ∧ 𝑦 = 𝑧) → 𝑥 = 𝑦) |
6 | 1, 5 | impbii 199 | 1 ⊢ (𝑥 = 𝑦 ↔ ∃𝑧(𝑥 = 𝑧 ∧ 𝑦 = 𝑧)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∃wex 1704 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 |
This theorem depends on definitions: df-bi 197 df-an 386 df-ex 1705 |
This theorem is referenced by: (None) |
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