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Mirrors > Home > ILE Home > Th. List > cbvralw | GIF version |
Description: Rule used to change bound variables, using implicit substitution. Version of cbvral 2714 with a disjoint variable condition. Although we don't do so yet, we expect this disjoint variable condition will allow us to remove reliance on ax-i12 1518 and ax-bndl 1520 in the proof. (Contributed by NM, 31-Jul-2003.) (Revised by Gino Giotto, 10-Jan-2024.) |
Ref | Expression |
---|---|
cbvralw.1 | ⊢ Ⅎ𝑦𝜑 |
cbvralw.2 | ⊢ Ⅎ𝑥𝜓 |
cbvralw.3 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvralw | ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2332 | . 2 ⊢ Ⅎ𝑥𝐴 | |
2 | nfcv 2332 | . 2 ⊢ Ⅎ𝑦𝐴 | |
3 | cbvralw.1 | . 2 ⊢ Ⅎ𝑦𝜑 | |
4 | cbvralw.2 | . 2 ⊢ Ⅎ𝑥𝜓 | |
5 | cbvralw.3 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
6 | 1, 2, 3, 4, 5 | cbvralfw 2708 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 Ⅎwnf 1471 ∀wral 2468 |
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-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-nf 1472 df-sb 1774 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 |
This theorem is referenced by: pcmptdvds 12361 lgseisenlem2 14848 |
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