![]() |
Mathbox for BJ |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-cbvaldv | Structured version Visualization version GIF version |
Description: Version of cbvald 2402 with a disjoint variable condition, which does not require ax-13 2367. (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-cbvaldv | ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1910 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | bj-cbvaldv.1 | . 2 ⊢ Ⅎ𝑦𝜑 | |
3 | bj-cbvaldv.2 | . 2 ⊢ (𝜑 → Ⅎ𝑦𝜓) | |
4 | nfv 1910 | . . 3 ⊢ Ⅎ𝑥𝜒 | |
5 | 4 | a1i 11 | . 2 ⊢ (𝜑 → Ⅎ𝑥𝜒) |
6 | bj-cbvaldv.3 | . 2 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) | |
7 | 1, 2, 3, 5, 6 | bj-cbv2v 36269 | 1 ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1532 Ⅎwnf 1778 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-10 2130 ax-11 2147 ax-12 2167 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-ex 1775 df-nf 1779 |
This theorem is referenced by: bj-cbvexdv 36271 bj-cbvaldvav 36274 |
Copyright terms: Public domain | W3C validator |