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| Mirrors > Home > MPE Home > Th. List > cbval | 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 2377. Check out cbvalw 2034, cbvalvw 2035, cbvalv1 2343 for versions requiring fewer axioms. (Contributed by NM, 13-May-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| cbval.1 | ⊢ Ⅎ𝑦𝜑 |
| cbval.2 | ⊢ Ⅎ𝑥𝜓 |
| cbval.3 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
| Ref | Expression |
|---|---|
| cbval | ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cbval.1 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
| 2 | cbval.2 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
| 3 | cbval.3 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
| 4 | 3 | biimpd 229 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) |
| 5 | 1, 2, 4 | cbv3 2402 | . 2 ⊢ (∀𝑥𝜑 → ∀𝑦𝜓) |
| 6 | 3 | biimprd 248 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜓 → 𝜑)) |
| 7 | 6 | equcoms 2019 | . . 3 ⊢ (𝑦 = 𝑥 → (𝜓 → 𝜑)) |
| 8 | 2, 1, 7 | cbv3 2402 | . 2 ⊢ (∀𝑦𝜓 → ∀𝑥𝜑) |
| 9 | 5, 8 | impbii 209 | 1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1538 Ⅎwnf 1783 |
| 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 2377 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-nf 1784 |
| This theorem is referenced by: cbvex 2404 cbvalv 2405 cbval2 2416 sb8 2522 |
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