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Mirrors > Home > ILE Home > Th. List > cbvaldvaw | GIF version |
Description: Rule used to change the bound variable in a universal quantifier with implicit substitution. Deduction form. Version of cbvaldva 1940 with a disjoint variable condition. (Contributed by David Moews, 1-May-2017.) (Revised by GG, 10-Jan-2024.) (Revised by Wolf Lammen, 10-Feb-2024.) |
Ref | Expression |
---|---|
cbvaldvaw.1 | ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
cbvaldvaw | ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvaldvaw.1 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) | |
2 | 1 | ancoms 268 | . . . . 5 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → (𝜓 ↔ 𝜒)) |
3 | 2 | pm5.74da 443 | . . . 4 ⊢ (𝑥 = 𝑦 → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒))) |
4 | 3 | cbvalvw 1931 | . . 3 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ ∀𝑦(𝜑 → 𝜒)) |
5 | 19.21v 1884 | . . 3 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)) | |
6 | 19.21v 1884 | . . 3 ⊢ (∀𝑦(𝜑 → 𝜒) ↔ (𝜑 → ∀𝑦𝜒)) | |
7 | 4, 5, 6 | 3bitr3i 210 | . 2 ⊢ ((𝜑 → ∀𝑥𝜓) ↔ (𝜑 → ∀𝑦𝜒)) |
8 | 7 | pm5.74ri 181 | 1 ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∀wal 1362 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1458 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 |
This theorem depends on definitions: df-bi 117 df-nf 1472 |
This theorem is referenced by: cbval2vw 1944 |
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