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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 7986. (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 3063 | . 2 ⊢ ∀𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ∈ V |
4 | abrexex2g 7988 | . 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 2106 {cab 2712 ∀wral 3059 ∃wrex 3068 Vcvv 3478 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-v 3480 df-ss 3980 df-uni 4913 df-iun 4998 |
This theorem is referenced by: abexssex 7994 abexex 7995 oprabrexex2 8002 ab2rexex 8003 ab2rexex2 8004 |
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