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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 7288. (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 3073 | . 2 ⊢ ∀𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ∈ V |
4 | abrexex2g 7291 | . 2 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ∈ V) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ∈ V) | |
5 | 1, 3, 4 | mp2an 672 | 1 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2145 {cab 2757 ∀wral 3061 ∃wrex 3062 Vcvv 3351 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pr 5034 ax-un 7096 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 |
This theorem is referenced by: abexssex 7296 abexex 7298 oprabrexex2 7305 ab2rexex 7306 ab2rexex2 7307 |
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