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Mirrors > Home > ILE Home > Th. List > cbvrexsv | GIF version |
Description: Change bound variable by using a substitution. (Contributed by NM, 2-Mar-2008.) (Revised by Andrew Salmon, 11-Jul-2011.) |
Ref | Expression |
---|---|
cbvrexsv | ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1466 | . . 3 ⊢ Ⅎ𝑧𝜑 | |
2 | nfs1v 1863 | . . 3 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑 | |
3 | sbequ12 1701 | . . 3 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑)) | |
4 | 1, 2, 3 | cbvrex 2587 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑧 ∈ 𝐴 [𝑧 / 𝑥]𝜑) |
5 | nfv 1466 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
6 | 5 | nfsb 1870 | . . 3 ⊢ Ⅎ𝑦[𝑧 / 𝑥]𝜑 |
7 | nfv 1466 | . . 3 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑 | |
8 | sbequ 1768 | . . 3 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
9 | 6, 7, 8 | cbvrex 2587 | . 2 ⊢ (∃𝑧 ∈ 𝐴 [𝑧 / 𝑥]𝜑 ↔ ∃𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑) |
10 | 4, 9 | bitri 182 | 1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 103 [wsb 1692 ∃wrex 2360 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-nf 1395 df-sb 1693 df-cleq 2081 df-clel 2084 df-nfc 2217 df-rex 2365 |
This theorem is referenced by: rspesbca 2923 rexxpf 4583 |
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