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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-nfeel2 | Structured version Visualization version GIF version |
Description: Non-freeness in an equality. (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-nfeel2 | ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 ∈ 𝑧) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1830 | . 2 ⊢ Ⅎ𝑥 𝑡 ∈ 𝑧 | |
2 | eleq1 2676 | . 2 ⊢ (𝑡 = 𝑦 → (𝑡 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧)) | |
3 | 1, 2 | bj-dvelimv 32029 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 ∈ 𝑧) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1473 Ⅎwnf 1699 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-cleq 2603 df-clel 2606 |
This theorem is referenced by: bj-axc14nf 32031 |
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