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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 2337. (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 1843 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | cbvald.1 | . 2 ⊢ Ⅎ𝑦𝜑 | |
3 | cbvald.2 | . 2 ⊢ (𝜑 → Ⅎ𝑦𝜓) | |
4 | nfvd 1844 | . 2 ⊢ (𝜑 → Ⅎ𝑥𝜒) | |
5 | cbvald.3 | . 2 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) | |
6 | 1, 2, 3, 4, 5 | cbv2 2270 | 1 ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∀wal 1481 Ⅎwnf 1708 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-ex 1705 df-nf 1710 |
This theorem is referenced by: cbvexd 2278 cbvaldvaOLD 2282 axextnd 9413 axrepndlem1 9414 axunndlem1 9417 axpowndlem2 9420 axpowndlem3 9421 axpowndlem4 9422 axregndlem2 9425 axregnd 9426 axinfnd 9428 axacndlem5 9433 axacnd 9434 axextdist 31705 distel 31709 wl-sb8eut 33359 |
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