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Mirrors > Home > MPE Home > Th. List > cbv3h | Structured version Visualization version GIF version |
Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2367. Use the weaker cbv3hv 2332 if possible. (Contributed by NM, 8-Jun-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 12-May-2018.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cbv3h.1 | ⊢ (𝜑 → ∀𝑦𝜑) |
cbv3h.2 | ⊢ (𝜓 → ∀𝑥𝜓) |
cbv3h.3 | ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) |
Ref | Expression |
---|---|
cbv3h | ⊢ (∀𝑥𝜑 → ∀𝑦𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbv3h.1 | . . 3 ⊢ (𝜑 → ∀𝑦𝜑) | |
2 | 1 | nf5i 2135 | . 2 ⊢ Ⅎ𝑦𝜑 |
3 | cbv3h.2 | . . 3 ⊢ (𝜓 → ∀𝑥𝜓) | |
4 | 3 | nf5i 2135 | . 2 ⊢ Ⅎ𝑥𝜓 |
5 | cbv3h.3 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) | |
6 | 2, 4, 5 | cbv3 2392 | 1 ⊢ (∀𝑥𝜑 → ∀𝑦𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1532 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-10 2130 ax-11 2147 ax-12 2167 ax-13 2367 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1775 df-nf 1779 |
This theorem is referenced by: (None) |
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