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Mirrors > Home > MPE Home > Th. List > cbvaldva | Structured version Visualization version GIF version |
Description: Rule used to change the bound variable in a universal quantifier with implicit substitution. Deduction form. See also cbvaldvaw 2036. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
cbvaldva.1 | ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
cbvaldva | ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1906 | . 2 ⊢ Ⅎ𝑦𝜑 | |
2 | nfvd 1907 | . 2 ⊢ (𝜑 → Ⅎ𝑦𝜓) | |
3 | cbvaldva.1 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) | |
4 | 3 | ex 413 | . 2 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) |
5 | 1, 2, 4 | cbvald 2419 | 1 ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∀wal 1526 |
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-11 2151 ax-12 2167 ax-13 2381 |
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: cbval2vv 2426 |
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