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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-cbvexdv | Structured version Visualization version GIF version |
Description: Version of cbvexd 2420 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-cbvaldv.1 | ⊢ Ⅎ𝑦𝜑 |
bj-cbvaldv.2 | ⊢ (𝜑 → Ⅎ𝑦𝜓) |
bj-cbvaldv.3 | ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) |
Ref | Expression |
---|---|
bj-cbvexdv | ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-cbvaldv.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
2 | bj-cbvaldv.2 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑦𝜓) | |
3 | 2 | nfnd 1849 | . . . 4 ⊢ (𝜑 → Ⅎ𝑦 ¬ 𝜓) |
4 | bj-cbvaldv.3 | . . . . 5 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) | |
5 | notbi 320 | . . . . 5 ⊢ ((𝜓 ↔ 𝜒) ↔ (¬ 𝜓 ↔ ¬ 𝜒)) | |
6 | 4, 5 | syl6ib 252 | . . . 4 ⊢ (𝜑 → (𝑥 = 𝑦 → (¬ 𝜓 ↔ ¬ 𝜒))) |
7 | 1, 3, 6 | bj-cbvaldv 34018 | . . 3 ⊢ (𝜑 → (∀𝑥 ¬ 𝜓 ↔ ∀𝑦 ¬ 𝜒)) |
8 | 7 | notbid 319 | . 2 ⊢ (𝜑 → (¬ ∀𝑥 ¬ 𝜓 ↔ ¬ ∀𝑦 ¬ 𝜒)) |
9 | df-ex 1772 | . 2 ⊢ (∃𝑥𝜓 ↔ ¬ ∀𝑥 ¬ 𝜓) | |
10 | df-ex 1772 | . 2 ⊢ (∃𝑦𝜒 ↔ ¬ ∀𝑦 ¬ 𝜒) | |
11 | 8, 9, 10 | 3bitr4g 315 | 1 ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∀wal 1526 ∃wex 1771 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 |
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-cbvexdvav 34023 |
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