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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 2032 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 2018 | . 2 ⊢ ∃𝑧 𝑦 = 𝑧 | |
2 | equtr 2024 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑦 = 𝑧 → 𝑥 = 𝑧)) | |
3 | 2 | ancrd 552 | . . 3 ⊢ (𝑥 = 𝑦 → (𝑦 = 𝑧 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑧))) |
4 | 3 | eximdv 1920 | . 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 396 ∃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 1913 ax-6 1971 ax-7 2011 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1782 |
This theorem is referenced by: sbequ2 2241 ax13lem1 2372 nfeqf 2379 wl-ax13lem1 36079 |
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