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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 7896. (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 3069 | . 2 ⊢ ∀𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ∈ V |
4 | abrexex2g 7898 | . 2 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ∈ V) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ∈ V) | |
5 | 1, 3, 4 | mp2an 691 | 1 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2107 {cab 2714 ∀wral 3065 ∃wrex 3074 Vcvv 3446 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ral 3066 df-rex 3075 df-v 3448 df-in 3918 df-ss 3928 df-uni 4867 df-iun 4957 |
This theorem is referenced by: abexssex 7904 abexex 7905 oprabrexex2 7912 ab2rexex 7913 ab2rexex2 7914 |
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