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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-cbvaldvav | Structured version Visualization version GIF version |
Description: Version of cbvaldva 2411 with a disjoint variable condition, which does not require ax-13 2374. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-cbvaldvav.1 | ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
bj-cbvaldvav | ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1921 | . 2 ⊢ Ⅎ𝑦𝜑 | |
2 | nfvd 1922 | . 2 ⊢ (𝜑 → Ⅎ𝑦𝜓) | |
3 | bj-cbvaldvav.1 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) | |
4 | 3 | ex 413 | . 2 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) |
5 | 1, 2, 4 | bj-cbvaldv 34990 | 1 ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∀wal 1540 |
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 2015 ax-10 2141 ax-11 2158 ax-12 2175 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-ex 1787 df-nf 1791 |
This theorem is referenced by: (None) |
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