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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-cbvexdv | Structured version Visualization version GIF version |
Description: Version of cbvexd 2373 with a disjoint variable condition, which does not require ax-13 2334. (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 1903 | . . . 4 ⊢ (𝜑 → Ⅎ𝑦 ¬ 𝜓) |
4 | bj-cbvaldv.3 | . . . . 5 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) | |
5 | notbi 311 | . . . . 5 ⊢ ((𝜓 ↔ 𝜒) ↔ (¬ 𝜓 ↔ ¬ 𝜒)) | |
6 | 4, 5 | syl6ib 243 | . . . 4 ⊢ (𝜑 → (𝑥 = 𝑦 → (¬ 𝜓 ↔ ¬ 𝜒))) |
7 | 1, 3, 6 | bj-cbvaldv 33320 | . . 3 ⊢ (𝜑 → (∀𝑥 ¬ 𝜓 ↔ ∀𝑦 ¬ 𝜒)) |
8 | 7 | notbid 310 | . 2 ⊢ (𝜑 → (¬ ∀𝑥 ¬ 𝜓 ↔ ¬ ∀𝑦 ¬ 𝜒)) |
9 | df-ex 1824 | . 2 ⊢ (∃𝑥𝜓 ↔ ¬ ∀𝑥 ¬ 𝜓) | |
10 | df-ex 1824 | . 2 ⊢ (∃𝑦𝜒 ↔ ¬ ∀𝑦 ¬ 𝜒) | |
11 | 8, 9, 10 | 3bitr4g 306 | 1 ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∀wal 1599 ∃wex 1823 Ⅎwnf 1827 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-10 2135 ax-11 2150 ax-12 2163 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-ex 1824 df-nf 1828 |
This theorem is referenced by: bj-cbvexdvav 33327 |
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