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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-dvelimdv1 | Structured version Visualization version GIF version |
Description: Curried form (exported form) of bj-dvelimdv 32027. (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-dvelimdv.nf | ⊢ Ⅎ𝑥𝜑 |
bj-dvelimdv.nf1 | ⊢ (𝜑 → Ⅎ𝑥𝜒) |
bj-dvelimdv.is | ⊢ (𝑧 = 𝑦 → (𝜒 ↔ 𝜓)) |
Ref | Expression |
---|---|
bj-dvelimdv1 | ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfeqf2 2285 | . . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦) | |
2 | bj-dvelimdv.nf1 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜒) | |
3 | bj-nfimt 32025 | . . . 4 ⊢ (Ⅎ𝑥 𝑧 = 𝑦 → (Ⅎ𝑥𝜒 → Ⅎ𝑥(𝑧 = 𝑦 → 𝜒))) | |
4 | 1, 2, 3 | syl2imc 40 | . . 3 ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥(𝑧 = 𝑦 → 𝜒))) |
5 | 4 | alrimdv 1844 | . 2 ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧Ⅎ𝑥(𝑧 = 𝑦 → 𝜒))) |
6 | bj-nfalt 31889 | . 2 ⊢ (∀𝑧Ⅎ𝑥(𝑧 = 𝑦 → 𝜒) → Ⅎ𝑥∀𝑧(𝑧 = 𝑦 → 𝜒)) | |
7 | bj-dvelimdv.is | . . . 4 ⊢ (𝑧 = 𝑦 → (𝜒 ↔ 𝜓)) | |
8 | 7 | equsalvw 1918 | . . 3 ⊢ (∀𝑧(𝑧 = 𝑦 → 𝜒) ↔ 𝜓) |
9 | 8 | nfbii 1770 | . 2 ⊢ (Ⅎ𝑥∀𝑧(𝑧 = 𝑦 → 𝜒) ↔ Ⅎ𝑥𝜓) |
10 | 5, 6, 9 | bj-syl66ib 32023 | 1 ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∀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 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 |
This theorem is referenced by: bj-dvelimv 32029 |
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