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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. (Contributed by NM, 13-May-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) |
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 221 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) |
5 | 1, 2, 4 | cbv3 2418 | . 2 ⊢ (∀𝑥𝜑 → ∀𝑦𝜓) |
6 | 3 | biimprd 240 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜓 → 𝜑)) |
7 | 6 | equcoms 2126 | . . 3 ⊢ (𝑦 = 𝑥 → (𝜓 → 𝜑)) |
8 | 2, 1, 7 | cbv3 2418 | . 2 ⊢ (∀𝑦𝜓 → ∀𝑥𝜑) |
9 | 5, 8 | impbii 201 | 1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∀wal 1656 Ⅎwnf 1884 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 |
This theorem depends on definitions: df-bi 199 df-an 387 df-ex 1881 df-nf 1885 |
This theorem is referenced by: cbvex 2425 cbval2 2430 sb8 2556 |
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