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Mirrors > Home > ILE Home > Th. List > sbcreug | GIF version |
Description: Interchange class substitution and restricted uniqueness quantifier. (Contributed by NM, 24-Feb-2013.) |
Ref | Expression |
---|---|
sbcreug | ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∃!𝑦 ∈ 𝐵 𝜑 ↔ ∃!𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsbcq2 2819 | . 2 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]∃!𝑦 ∈ 𝐵 𝜑 ↔ [𝐴 / 𝑥]∃!𝑦 ∈ 𝐵 𝜑)) | |
2 | dfsbcq2 2819 | . . 3 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
3 | 2 | reubidv 2538 | . 2 ⊢ (𝑧 = 𝐴 → (∃!𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑 ↔ ∃!𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑)) |
4 | nfcv 2220 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
5 | nfs1v 1857 | . . . 4 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑 | |
6 | 4, 5 | nfreuxy 2529 | . . 3 ⊢ Ⅎ𝑥∃!𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑 |
7 | sbequ12 1695 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑)) | |
8 | 7 | reubidv 2538 | . . 3 ⊢ (𝑥 = 𝑧 → (∃!𝑦 ∈ 𝐵 𝜑 ↔ ∃!𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑)) |
9 | 6, 8 | sbie 1715 | . 2 ⊢ ([𝑧 / 𝑥]∃!𝑦 ∈ 𝐵 𝜑 ↔ ∃!𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑) |
10 | 1, 3, 9 | vtoclbg 2660 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∃!𝑦 ∈ 𝐵 𝜑 ↔ ∃!𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 = wceq 1285 ∈ wcel 1434 [wsb 1686 ∃!wreu 2351 [wsbc 2816 |
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 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2064 |
This theorem depends on definitions: df-bi 115 df-tru 1288 df-nf 1391 df-sb 1687 df-eu 1945 df-clab 2069 df-cleq 2075 df-clel 2078 df-nfc 2209 df-reu 2356 df-v 2604 df-sbc 2817 |
This theorem is referenced by: (None) |
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