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Mirrors > Home > MPE Home > Th. List > sbcex2 | Structured version Visualization version GIF version |
Description: Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.) (Revised by NM, 18-Aug-2018.) |
Ref | Expression |
---|---|
sbcex2 | ⊢ ([𝐴 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcex 3779 | . 2 ⊢ ([𝐴 / 𝑦]∃𝑥𝜑 → 𝐴 ∈ V) | |
2 | sbcex 3779 | . . 3 ⊢ ([𝐴 / 𝑦]𝜑 → 𝐴 ∈ V) | |
3 | 2 | exlimiv 1922 | . 2 ⊢ (∃𝑥[𝐴 / 𝑦]𝜑 → 𝐴 ∈ V) |
4 | dfsbcq2 3772 | . . 3 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑦]∃𝑥𝜑 ↔ [𝐴 / 𝑦]∃𝑥𝜑)) | |
5 | dfsbcq2 3772 | . . . 4 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑦]𝜑 ↔ [𝐴 / 𝑦]𝜑)) | |
6 | 5 | exbidv 1913 | . . 3 ⊢ (𝑧 = 𝐴 → (∃𝑥[𝑧 / 𝑦]𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑)) |
7 | sbex 2279 | . . 3 ⊢ ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑) | |
8 | 4, 6, 7 | vtoclbg 3566 | . 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑)) |
9 | 1, 3, 8 | pm5.21nii 380 | 1 ⊢ ([𝐴 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 = wceq 1528 ∃wex 1771 [wsb 2060 ∈ wcel 2105 Vcvv 3492 [wsbc 3769 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-v 3494 df-sbc 3770 |
This theorem is referenced by: sbcabel 3858 csbuni 4858 csbxp 5643 csbdm 5759 sbcfung 6372 bnj89 31890 bnj985 32124 csbwrecsg 34490 csboprabg 34493 sbcexf 35274 onfrALTlem5 40753 onfrALTlem5VD 41096 csbxpgVD 41105 csbrngVD 41107 csbunigVD 41109 |
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