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| Description: Existence of an existentially restricted class abstraction. 𝜑 normally has free-variable parameters 𝑥 and 𝑦. See also abrexex 7988. (Contributed by NM, 12-Sep-2004.) | 
| Ref | Expression | 
|---|---|
| abrexex2.1 | ⊢ 𝐴 ∈ V | 
| abrexex2.2 | ⊢ {𝑦 ∣ 𝜑} ∈ V | 
| Ref | Expression | 
|---|---|
| abrexex2 | ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ∈ V | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | abrexex2.1 | . 2 ⊢ 𝐴 ∈ V | |
| 2 | abrexex2.2 | . . 3 ⊢ {𝑦 ∣ 𝜑} ∈ V | |
| 3 | 2 | rgenw 3064 | . 2 ⊢ ∀𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ∈ V | 
| 4 | abrexex2g 7990 | . 2 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ∈ V) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ∈ V) | |
| 5 | 1, 3, 4 | mp2an 692 | 1 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ∈ V | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∈ wcel 2107 {cab 2713 ∀wral 3060 ∃wrex 3069 Vcvv 3479 | 
| 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-rep 5278 ax-sep 5295 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-v 3481 df-ss 3967 df-uni 4907 df-iun 4992 | 
| This theorem is referenced by: abexssex 7996 abexex 7997 oprabrexex2 8004 ab2rexex 8005 ab2rexex2 8006 | 
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