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Mirrors > Home > MPE Home > Th. List > cbv3 | Structured version Visualization version GIF version |
Description: Rule used to change bound variables, using implicit substitution, that does not use ax-c9 33193. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-May-2018.) |
Ref | Expression |
---|---|
cbv3.1 | ⊢ Ⅎ𝑦𝜑 |
cbv3.2 | ⊢ Ⅎ𝑥𝜓 |
cbv3.3 | ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) |
Ref | Expression |
---|---|
cbv3 | ⊢ (∀𝑥𝜑 → ∀𝑦𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbv3.1 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
2 | 1 | nfal 2139 | . 2 ⊢ Ⅎ𝑦∀𝑥𝜑 |
3 | cbv3.2 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
4 | cbv3.3 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) | |
5 | 3, 4 | spim 2242 | . 2 ⊢ (∀𝑥𝜑 → 𝜓) |
6 | 2, 5 | alrimi 2069 | 1 ⊢ (∀𝑥𝜑 → ∀𝑦𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀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-an 385 df-ex 1696 df-nf 1701 |
This theorem is referenced by: cbv3h 2254 cbv1 2255 cbval 2259 axc16i 2310 bj-mo3OLD 32022 |
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