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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. See cbvalvw 2086 for a version requiring fewer axioms, to be preferred when sufficient. (Contributed by NM, 5-Aug-1993.) Remove dependency on ax-10 2135. (Revised by Wolf Lammen, 17-Jul-2021.) |
Ref | Expression |
---|---|
cbvalv.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvalv | ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-5 1953 | . . . 4 ⊢ (𝜑 → ∀𝑦𝜑) | |
2 | 1 | hbal 2160 | . . 3 ⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑) |
3 | cbvalv.1 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
4 | 3 | spv 2358 | . . 3 ⊢ (∀𝑥𝜑 → 𝜓) |
5 | 2, 4 | alrimih 1867 | . 2 ⊢ (∀𝑥𝜑 → ∀𝑦𝜓) |
6 | ax-5 1953 | . . . 4 ⊢ (𝜓 → ∀𝑥𝜓) | |
7 | 6 | hbal 2160 | . . 3 ⊢ (∀𝑦𝜓 → ∀𝑥∀𝑦𝜓) |
8 | 3 | equcoms 2067 | . . . . 5 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓)) |
9 | 8 | bicomd 215 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝜓 ↔ 𝜑)) |
10 | 9 | spv 2358 | . . 3 ⊢ (∀𝑦𝜓 → 𝜑) |
11 | 7, 10 | alrimih 1867 | . 2 ⊢ (∀𝑦𝜓 → ∀𝑥𝜑) |
12 | 5, 11 | impbii 201 | 1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∀wal 1599 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-11 2150 ax-12 2163 ax-13 2334 |
This theorem depends on definitions: df-bi 199 df-an 387 df-ex 1824 df-nf 1828 |
This theorem is referenced by: cbvexv 2371 cbvaldva 2376 cbval2v 2378 cdeqal1 3643 |
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