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| Description: The intersection of a nonempty class abstraction exists. (Contributed by NM, 21-Oct-2003.) | 
| Ref | Expression | 
|---|---|
| intexab | ⊢ (∃𝑥𝜑 ↔ ∩ {𝑥 ∣ 𝜑} ∈ V) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | abn0 4384 | . 2 ⊢ ({𝑥 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥𝜑) | |
| 2 | intex 5343 | . 2 ⊢ ({𝑥 ∣ 𝜑} ≠ ∅ ↔ ∩ {𝑥 ∣ 𝜑} ∈ V) | |
| 3 | 1, 2 | bitr3i 277 | 1 ⊢ (∃𝑥𝜑 ↔ ∩ {𝑥 ∣ 𝜑} ∈ V) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∃wex 1778 ∈ wcel 2107 {cab 2713 ≠ wne 2939 Vcvv 3479 ∅c0 4332 ∩ cint 4945 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-in 3957 df-ss 3967 df-nul 4333 df-int 4946 | 
| This theorem is referenced by: intexrab 5346 tcmin 9782 cfval 10288 efgval 19736 relintabex 43599 rclexi 43633 rtrclex 43635 trclexi 43638 rtrclexi 43639 aiotaexb 47106 | 
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