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| Description: The intersection of a nonempty restricted class abstraction exists. (Contributed by NM, 21-Oct-2003.) | 
| Ref | Expression | 
|---|---|
| intexrab | ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∩ {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | intexab 5346 | . 2 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∩ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∈ V) | |
| 2 | df-rex 3071 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
| 3 | df-rab 3437 | . . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
| 4 | 3 | inteqi 4950 | . . 3 ⊢ ∩ {𝑥 ∈ 𝐴 ∣ 𝜑} = ∩ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | 
| 5 | 4 | eleq1i 2832 | . 2 ⊢ (∩ {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V ↔ ∩ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∈ V) | 
| 6 | 1, 2, 5 | 3bitr4i 303 | 1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∩ {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 ∃wex 1779 ∈ wcel 2108 {cab 2714 ∃wrex 3070 {crab 3436 Vcvv 3480 ∩ cint 4946 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-in 3958 df-ss 3968 df-nul 4334 df-int 4947 | 
| This theorem is referenced by: onintrab2 7817 rankf 9834 rankvalb 9837 cardf2 9983 tskmval 10879 rgspnval 20612 lspval 20973 aspval 21893 clsval 23045 spanval 31352 fldgenval 33314 | 
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