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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-axc14nf | Structured version Visualization version GIF version | ||
| Description: Proof of a version of axc14 2468 using the "nonfree" idiom. (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.) |
| Ref | Expression |
|---|---|
| bj-axc14nf | ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧 𝑥 ∈ 𝑦)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bj-nfeel2 36855 | . 2 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → Ⅎ𝑧 𝑥 ∈ 𝑡) | |
| 2 | elequ2 2123 | . 2 ⊢ (𝑡 = 𝑦 → (𝑥 ∈ 𝑡 ↔ 𝑥 ∈ 𝑦)) | |
| 3 | 1, 2 | bj-dvelimdv1 36853 | 1 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧 𝑥 ∈ 𝑦)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∀wal 1538 Ⅎwnf 1783 |
| 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 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-13 2377 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 |
| This theorem is referenced by: bj-axc14 36857 |
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