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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-ax6elem2 | Structured version Visualization version GIF version | ||
| Description: Lemma for bj-ax6e 36669. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) |
| Ref | Expression |
|---|---|
| bj-ax6elem2 | ⊢ (∀𝑥 𝑦 = 𝑧 → ∃𝑥 𝑥 = 𝑦) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ax6ev 1969 | . . 3 ⊢ ∃𝑥 𝑥 = 𝑧 | |
| 2 | equeucl 2023 | . . 3 ⊢ (𝑥 = 𝑧 → (𝑦 = 𝑧 → 𝑥 = 𝑦)) | |
| 3 | 1, 2 | eximii 1837 | . 2 ⊢ ∃𝑥(𝑦 = 𝑧 → 𝑥 = 𝑦) |
| 4 | 3 | 19.35i 1878 | 1 ⊢ (∀𝑥 𝑦 = 𝑧 → ∃𝑥 𝑥 = 𝑦) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∀wal 1538 ∃wex 1779 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 |
| This theorem is referenced by: bj-ax6e 36669 |
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