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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-cleljusti | Structured version Visualization version GIF version |
Description: One direction of cleljust 2116, requiring only ax-1 6-- ax-5 1914 and ax8v1 2111. (Contributed by BJ, 31-Dec-2020.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-cleljusti | ⊢ (∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ 𝑦) → 𝑥 ∈ 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax8v1 2111 | . . 3 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝑦 → 𝑥 ∈ 𝑦)) | |
2 | 1 | imp 408 | . 2 ⊢ ((𝑧 = 𝑥 ∧ 𝑧 ∈ 𝑦) → 𝑥 ∈ 𝑦) |
3 | 2 | exlimiv 1934 | 1 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ 𝑦) → 𝑥 ∈ 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∃wex 1782 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-8 2109 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 |
This theorem is referenced by: (None) |
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