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Mirrors > Home > ILE Home > Th. List > cbvralv2 | GIF version |
Description: Rule used to change the bound variable in a restricted universal quantifier with implicit substitution which also changes the quantifier domain. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
cbvralv2.1 | ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)) |
cbvralv2.2 | ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
cbvralv2 | ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑦 ∈ 𝐵 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2178 | . 2 ⊢ Ⅎ𝑦𝐴 | |
2 | nfcv 2178 | . 2 ⊢ Ⅎ𝑥𝐵 | |
3 | nfv 1421 | . 2 ⊢ Ⅎ𝑦𝜓 | |
4 | nfv 1421 | . 2 ⊢ Ⅎ𝑥𝜒 | |
5 | cbvralv2.2 | . 2 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) | |
6 | cbvralv2.1 | . 2 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)) | |
7 | 1, 2, 3, 4, 5, 6 | cbvralcsf 2908 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑦 ∈ 𝐵 𝜒) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 = wceq 1243 ∀wral 2306 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-sbc 2765 df-csb 2853 |
This theorem is referenced by: (None) |
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