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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-dvelimdv1 | Structured version Visualization version GIF version |
Description: Curried (exported) form of bj-dvelimdv 34072 (of course, one is directly provable from the other, but we keep this proof for illustration purposes). (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-dvelimdv.nf | ⊢ (𝜑 → Ⅎ𝑥𝜒) |
bj-dvelimdv.is | ⊢ (𝑧 = 𝑦 → (𝜒 ↔ 𝜓)) |
Ref | Expression |
---|---|
bj-dvelimdv1 | ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfeqf2 2386 | . . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦) | |
2 | bj-dvelimdv.nf | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜒) | |
3 | bj-nfimt 33868 | . . . 4 ⊢ (Ⅎ𝑥 𝑧 = 𝑦 → (Ⅎ𝑥𝜒 → Ⅎ𝑥(𝑧 = 𝑦 → 𝜒))) | |
4 | 1, 2, 3 | syl2imc 41 | . . 3 ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥(𝑧 = 𝑦 → 𝜒))) |
5 | 4 | alrimdv 1921 | . 2 ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧Ⅎ𝑥(𝑧 = 𝑦 → 𝜒))) |
6 | bj-nfalt 33942 | . 2 ⊢ (∀𝑧Ⅎ𝑥(𝑧 = 𝑦 → 𝜒) → Ⅎ𝑥∀𝑧(𝑧 = 𝑦 → 𝜒)) | |
7 | bj-dvelimdv.is | . . . 4 ⊢ (𝑧 = 𝑦 → (𝜒 ↔ 𝜓)) | |
8 | 7 | equsalvw 2001 | . . 3 ⊢ (∀𝑧(𝑧 = 𝑦 → 𝜒) ↔ 𝜓) |
9 | 8 | nfbii 1843 | . 2 ⊢ (Ⅎ𝑥∀𝑧(𝑧 = 𝑦 → 𝜒) ↔ Ⅎ𝑥𝜓) |
10 | 5, 6, 9 | bj-syl66ib 33787 | 1 ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∀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-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-ex 1772 df-nf 1776 |
This theorem is referenced by: bj-dvelimv 34074 bj-axc14nf 34076 |
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