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Mirrors > Home > MPE Home > Th. List > cbvald | Structured version Visualization version GIF version |
Description: Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 2325. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) (Revised by Wolf Lammen, 13-May-2018.) |
Ref | Expression |
---|---|
cbvald.1 | ⊢ Ⅎ𝑦𝜑 |
cbvald.2 | ⊢ (𝜑 → Ⅎ𝑦𝜓) |
cbvald.3 | ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) |
Ref | Expression |
---|---|
cbvald | ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1830 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | cbvald.1 | . 2 ⊢ Ⅎ𝑦𝜑 | |
3 | cbvald.2 | . 2 ⊢ (𝜑 → Ⅎ𝑦𝜓) | |
4 | nfvd 1831 | . 2 ⊢ (𝜑 → Ⅎ𝑥𝜒) | |
5 | cbvald.3 | . 2 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) | |
6 | 1, 2, 3, 4, 5 | cbv2 2258 | 1 ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∀wal 1473 Ⅎwnf 1699 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-ex 1696 df-nf 1701 |
This theorem is referenced by: cbvexd 2266 cbvaldvaOLD 2270 axextnd 9292 axrepndlem1 9293 axunndlem1 9296 axpowndlem2 9299 axpowndlem3 9300 axpowndlem4 9301 axregndlem2 9304 axregnd 9305 axinfnd 9307 axacndlem5 9312 axacnd 9313 axextdist 30949 distel 30953 wl-sb8eut 32538 |
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