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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 2472. (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 2013 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | cbvald.1 | . 2 ⊢ Ⅎ𝑦𝜑 | |
3 | cbvald.2 | . 2 ⊢ (𝜑 → Ⅎ𝑦𝜓) | |
4 | nfvd 2014 | . 2 ⊢ (𝜑 → Ⅎ𝑥𝜒) | |
5 | cbvald.3 | . 2 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) | |
6 | 1, 2, 3, 4, 5 | cbv2 2422 | 1 ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∀wal 1654 Ⅎwnf 1882 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-ex 1879 df-nf 1883 |
This theorem is referenced by: cbvexd 2428 axextnd 9735 axrepndlem1 9736 axunndlem1 9739 axpowndlem2 9742 axpowndlem3 9743 axpowndlem4 9744 axregndlem2 9747 axregnd 9748 axinfnd 9750 axacndlem5 9755 axacnd 9756 axextdist 32238 distel 32242 wl-sb8eut 33898 |
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