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| Mirrors > Home > MPE Home > Th. List > equvinva | Structured version Visualization version GIF version | ||
| Description: A modified version of the forward implication of equvinv 2027 adapted to common usage. (Contributed by Wolf Lammen, 8-Sep-2018.) | 
| Ref | Expression | 
|---|---|
| equvinva | ⊢ (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧 ∧ 𝑦 = 𝑧)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ax6evr 2013 | . 2 ⊢ ∃𝑧 𝑦 = 𝑧 | |
| 2 | equtr 2019 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑦 = 𝑧 → 𝑥 = 𝑧)) | |
| 3 | 2 | ancrd 551 | . . 3 ⊢ (𝑥 = 𝑦 → (𝑦 = 𝑧 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑧))) | 
| 4 | 3 | eximdv 1916 | . 2 ⊢ (𝑥 = 𝑦 → (∃𝑧 𝑦 = 𝑧 → ∃𝑧(𝑥 = 𝑧 ∧ 𝑦 = 𝑧))) | 
| 5 | 1, 4 | mpi 20 | 1 ⊢ (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧 ∧ 𝑦 = 𝑧)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∃wex 1778 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1779 | 
| This theorem is referenced by: sbequ2 2248 ax13lem1 2378 nfeqf 2385 wl-ax13lem1 37496 | 
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