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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 1539 | . . 3 ⊢ Ⅎ𝑧𝜑 | |
2 | nfs1v 1951 | . . 3 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑 | |
3 | sbequ12 1782 | . . 3 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑)) | |
4 | 1, 2, 3 | cbvrex 2715 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑧 ∈ 𝐴 [𝑧 / 𝑥]𝜑) |
5 | nfv 1539 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
6 | 5 | nfsb 1958 | . . 3 ⊢ Ⅎ𝑦[𝑧 / 𝑥]𝜑 |
7 | nfv 1539 | . . 3 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑 | |
8 | sbequ 1851 | . . 3 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
9 | 6, 7, 8 | cbvrex 2715 | . 2 ⊢ (∃𝑧 ∈ 𝐴 [𝑧 / 𝑥]𝜑 ↔ ∃𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑) |
10 | 4, 9 | bitri 184 | 1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 [wsb 1773 ∃wrex 2469 |
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-rex 2474 |
This theorem is referenced by: rspesbca 3062 rexxpf 4792 |
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