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Mirrors > Home > ILE Home > Th. List > cbvralfw | GIF version |
Description: Rule used to change bound variables, using implicit substitution. Version of cbvralf 2682 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 1494 and ax-bndl 1496 in the proof. (Contributed by NM, 7-Mar-2004.) (Revised by Gino Giotto, 23-May-2024.) |
Ref | Expression |
---|---|
cbvralfw.1 | ⊢ Ⅎ𝑥𝐴 |
cbvralfw.2 | ⊢ Ⅎ𝑦𝐴 |
cbvralfw.3 | ⊢ Ⅎ𝑦𝜑 |
cbvralfw.4 | ⊢ Ⅎ𝑥𝜓 |
cbvralfw.5 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvralfw | ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvralfw.2 | . . . . 5 ⊢ Ⅎ𝑦𝐴 | |
2 | 1 | nfcri 2300 | . . . 4 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴 |
3 | cbvralfw.3 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
4 | 2, 3 | nfim 1559 | . . 3 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 → 𝜑) |
5 | cbvralfw.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
6 | 5 | nfcri 2300 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴 |
7 | cbvralfw.4 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
8 | 6, 7 | nfim 1559 | . . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 → 𝜓) |
9 | eleq1w 2225 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
10 | cbvralfw.5 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
11 | 9, 10 | imbi12d 233 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 → 𝜑) ↔ (𝑦 ∈ 𝐴 → 𝜓))) |
12 | 4, 8, 11 | cbvalv1 1738 | . 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜓)) |
13 | df-ral 2447 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) | |
14 | df-ral 2447 | . 2 ⊢ (∀𝑦 ∈ 𝐴 𝜓 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜓)) | |
15 | 12, 13, 14 | 3bitr4i 211 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1340 Ⅎwnf 1447 ∈ wcel 2135 Ⅎwnfc 2293 ∀wral 2442 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-nf 1448 df-sb 1750 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 |
This theorem is referenced by: cbvralw 2684 |
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