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Mirrors > Home > MPE Home > Th. List > cbv2 | Structured version Visualization version GIF version |
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) Format hypotheses to common style. (Revised by Wolf Lammen, 13-May-2018.) |
Ref | Expression |
---|---|
cbv2.1 | ⊢ Ⅎ𝑥𝜑 |
cbv2.2 | ⊢ Ⅎ𝑦𝜑 |
cbv2.3 | ⊢ (𝜑 → Ⅎ𝑦𝜓) |
cbv2.4 | ⊢ (𝜑 → Ⅎ𝑥𝜒) |
cbv2.5 | ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) |
Ref | Expression |
---|---|
cbv2 | ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbv2.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | cbv2.2 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
3 | 2 | nf5ri 2053 | . . 3 ⊢ (𝜑 → ∀𝑦𝜑) |
4 | 1, 3 | alrimi 2069 | . 2 ⊢ (𝜑 → ∀𝑥∀𝑦𝜑) |
5 | cbv2.3 | . . . 4 ⊢ (𝜑 → Ⅎ𝑦𝜓) | |
6 | 5 | nf5rd 2054 | . . 3 ⊢ (𝜑 → (𝜓 → ∀𝑦𝜓)) |
7 | cbv2.4 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜒) | |
8 | 7 | nf5rd 2054 | . . 3 ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒)) |
9 | cbv2.5 | . . 3 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) | |
10 | 6, 8, 9 | cbv2h 2257 | . 2 ⊢ (∀𝑥∀𝑦𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) |
11 | 4, 10 | syl 17 | 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: cbvald 2265 sb9 2414 wl-cbvalnaed 32498 wl-sb8t 32512 |
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