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Mirrors > Home > MPE Home > Th. List > abrexex2 | Structured version Visualization version GIF version |
Description: Existence of an existentially restricted class abstraction. 𝜑 normally has free-variable parameters 𝑥 and 𝑦. See also abrexex 7967. (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 3054 | . 2 ⊢ ∀𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ∈ V |
4 | abrexex2g 7969 | . 2 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ∈ V) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ∈ V) | |
5 | 1, 3, 4 | mp2an 690 | 1 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2098 {cab 2702 ∀wral 3050 ∃wrex 3059 Vcvv 3461 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ral 3051 df-rex 3060 df-v 3463 df-ss 3961 df-uni 4910 df-iun 4999 |
This theorem is referenced by: abexssex 7975 abexex 7976 oprabrexex2 7983 ab2rexex 7984 ab2rexex2 7985 |
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