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Mirrors > Home > ILE Home > Th. List > cbvral2v | GIF version |
Description: Change bound variables of double restricted universal quantification, using implicit substitution. (Contributed by NM, 10-Aug-2004.) |
Ref | Expression |
---|---|
cbvral2v.1 | ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜒)) |
cbvral2v.2 | ⊢ (𝑦 = 𝑤 → (𝜒 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvral2v | ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐵 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvral2v.1 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜒)) | |
2 | 1 | ralbidv 2326 | . . 3 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 𝜒)) |
3 | 2 | cbvralv 2533 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑧 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒) |
4 | cbvral2v.2 | . . . 4 ⊢ (𝑦 = 𝑤 → (𝜒 ↔ 𝜓)) | |
5 | 4 | cbvralv 2533 | . . 3 ⊢ (∀𝑦 ∈ 𝐵 𝜒 ↔ ∀𝑤 ∈ 𝐵 𝜓) |
6 | 5 | ralbii 2330 | . 2 ⊢ (∀𝑧 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒 ↔ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐵 𝜓) |
7 | 3, 6 | bitri 173 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐵 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 ∀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-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 |
This theorem is referenced by: cbvral3v 2543 fununi 4967 cauappcvgprlemlim 6759 caucvgprlemnkj 6764 caucvgprlemcl 6774 caucvgprprlemcbv 6785 axcaucvglemcau 6972 iseqdistr 9249 |
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