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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-cbvaldv | Structured version Visualization version GIF version |
Description: Version of cbvald 2407 with a disjoint variable condition, which does not require ax-13 2372. (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 1921 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | bj-cbvaldv.1 | . 2 ⊢ Ⅎ𝑦𝜑 | |
3 | bj-cbvaldv.2 | . 2 ⊢ (𝜑 → Ⅎ𝑦𝜓) | |
4 | nfv 1921 | . . 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 34611 | 1 ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∀wal 1540 Ⅎwnf 1790 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-10 2145 ax-11 2162 ax-12 2179 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-ex 1787 df-nf 1791 |
This theorem is referenced by: bj-cbvexdv 34613 bj-cbvaldvav 34616 |
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