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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 2478 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 34075 | . 2 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → Ⅎ𝑧 𝑥 ∈ 𝑡) | |
2 | elequ2 2120 | . 2 ⊢ (𝑡 = 𝑦 → (𝑥 ∈ 𝑡 ↔ 𝑥 ∈ 𝑦)) | |
3 | 1, 2 | bj-dvelimdv1 34073 | 1 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧 𝑥 ∈ 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1526 Ⅎwnf 1775 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-13 2381 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 |
This theorem is referenced by: bj-axc14 34077 |
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