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Mirrors > Home > ILE Home > Th. List > sbcex2 | GIF version |
Description: Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.) |
Ref | Expression |
---|---|
sbcex2 | ⊢ ([𝐴 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcex 2772 | . 2 ⊢ ([𝐴 / 𝑦]∃𝑥𝜑 → 𝐴 ∈ V) | |
2 | sbcex 2772 | . . 3 ⊢ ([𝐴 / 𝑦]𝜑 → 𝐴 ∈ V) | |
3 | 2 | exlimiv 1489 | . 2 ⊢ (∃𝑥[𝐴 / 𝑦]𝜑 → 𝐴 ∈ V) |
4 | dfsbcq2 2767 | . . 3 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑦]∃𝑥𝜑 ↔ [𝐴 / 𝑦]∃𝑥𝜑)) | |
5 | dfsbcq2 2767 | . . . 4 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑦]𝜑 ↔ [𝐴 / 𝑦]𝜑)) | |
6 | 5 | exbidv 1706 | . . 3 ⊢ (𝑧 = 𝐴 → (∃𝑥[𝑧 / 𝑦]𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑)) |
7 | sbex 1880 | . . 3 ⊢ ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑) | |
8 | 4, 6, 7 | vtoclbg 2614 | . 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑)) |
9 | 1, 3, 8 | pm5.21nii 620 | 1 ⊢ ([𝐴 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 = wceq 1243 ∃wex 1381 ∈ wcel 1393 [wsb 1645 Vcvv 2557 [wsbc 2764 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-sbc 2765 |
This theorem is referenced by: csbdmg 4529 |
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