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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 3782 | . 2 ⊢ ([𝐴 / 𝑦]∃𝑥𝜑 → 𝐴 ∈ V) | |
2 | sbcex 3782 | . . 3 ⊢ ([𝐴 / 𝑦]𝜑 → 𝐴 ∈ V) | |
3 | 2 | exlimiv 1931 | . 2 ⊢ (∃𝑥[𝐴 / 𝑦]𝜑 → 𝐴 ∈ V) |
4 | dfsbcq2 3775 | . . 3 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑦]∃𝑥𝜑 ↔ [𝐴 / 𝑦]∃𝑥𝜑)) | |
5 | dfsbcq2 3775 | . . . 4 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑦]𝜑 ↔ [𝐴 / 𝑦]𝜑)) | |
6 | 5 | exbidv 1922 | . . 3 ⊢ (𝑧 = 𝐴 → (∃𝑥[𝑧 / 𝑦]𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑)) |
7 | sbex 2288 | . . 3 ⊢ ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑) | |
8 | 4, 6, 7 | vtoclbg 3569 | . 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑)) |
9 | 1, 3, 8 | pm5.21nii 382 | 1 ⊢ ([𝐴 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 ∃wex 1780 [wsb 2069 ∈ wcel 2114 Vcvv 3494 [wsbc 3772 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-v 3496 df-sbc 3773 |
This theorem is referenced by: sbcabel 3861 csbuni 4867 csbxp 5650 csbdm 5766 sbcfung 6379 bnj89 31991 bnj985v 32225 bnj985 32226 csbwrecsg 34611 csboprabg 34614 sbcexf 35408 onfrALTlem5 40896 onfrALTlem5VD 41239 csbxpgVD 41248 csbrngVD 41250 csbunigVD 41252 |
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