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| Mirrors > Home > MPE Home > Th. List > cbvalv | Structured version Visualization version GIF version | ||
| Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2376. See cbvalvw 2035 for a version requiring fewer axioms, to be preferred when sufficient. (Contributed by NM, 5-Aug-1993.) Remove dependency on ax-10 2141 and shorten proof. (Revised by Wolf Lammen, 11-Sep-2023.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| cbvalv.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
| Ref | Expression |
|---|---|
| cbvalv | ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nfv 1914 | . 2 ⊢ Ⅎ𝑦𝜑 | |
| 2 | nfv 1914 | . 2 ⊢ Ⅎ𝑥𝜓 | |
| 3 | cbvalv.1 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
| 4 | 1, 2, 3 | cbval 2402 | 1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1538 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-11 2157 ax-12 2177 ax-13 2376 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-nf 1784 |
| This theorem is referenced by: cbval2vv 2417 cdeqal1 3776 |
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