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Mirrors > Home > MPE Home > Th. List > elrabf | Structured version Visualization version GIF version |
Description: Membership in a restricted class abstraction, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable restrictions. (Contributed by NM, 21-Sep-2003.) |
Ref | Expression |
---|---|
elrabf.1 | ⊢ Ⅎ𝑥𝐴 |
elrabf.2 | ⊢ Ⅎ𝑥𝐵 |
elrabf.3 | ⊢ Ⅎ𝑥𝜓 |
elrabf.4 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
elrabf | ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ (𝐴 ∈ 𝐵 ∧ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3429 | . 2 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} → 𝐴 ∈ V) | |
2 | elex 3429 | . . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) | |
3 | 2 | adantr 474 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → 𝐴 ∈ V) |
4 | df-rab 3126 | . . . 4 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} | |
5 | 4 | eleq2i 2898 | . . 3 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ 𝐴 ∈ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)}) |
6 | elrabf.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
7 | elrabf.2 | . . . . . 6 ⊢ Ⅎ𝑥𝐵 | |
8 | 6, 7 | nfel 2982 | . . . . 5 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵 |
9 | elrabf.3 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
10 | 8, 9 | nfan 2002 | . . . 4 ⊢ Ⅎ𝑥(𝐴 ∈ 𝐵 ∧ 𝜓) |
11 | eleq1 2894 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) | |
12 | elrabf.4 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
13 | 11, 12 | anbi12d 624 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 ∧ 𝜑) ↔ (𝐴 ∈ 𝐵 ∧ 𝜓))) |
14 | 6, 10, 13 | elabgf 3567 | . . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} ↔ (𝐴 ∈ 𝐵 ∧ 𝜓))) |
15 | 5, 14 | syl5bb 275 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ (𝐴 ∈ 𝐵 ∧ 𝜓))) |
16 | 1, 3, 15 | pm5.21nii 370 | 1 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ (𝐴 ∈ 𝐵 ∧ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1656 Ⅎwnf 1882 ∈ wcel 2164 {cab 2811 Ⅎwnfc 2956 {crab 3121 Vcvv 3414 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-ext 2803 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-rab 3126 df-v 3416 |
This theorem is referenced by: rabtru 3582 elrabOLD 3586 invdisjrab 4860 rabxfrd 5117 f1ossf1o 6645 onminsb 7260 nnawordex 7984 tskwe 9089 rabssnn0fi 13080 iundisj 23714 iundisjf 29938 iundisjfi 30091 bnj1388 31636 sltval2 32337 phpreu 33929 poimirlem26 33972 rfcnpre3 40003 rfcnpre4 40004 uzwo4 40031 disjinfi 40181 allbutfiinf 40435 fsumiunss 40595 fnlimfvre 40694 stoweidlem26 41030 stoweidlem27 41031 stoweidlem31 41035 stoweidlem34 41038 stoweidlem51 41055 stoweidlem52 41056 stoweidlem59 41063 fourierdlem20 41131 fourierdlem79 41189 pimdecfgtioc 41712 smfpimcclem 41800 prmdvdsfmtnof1lem1 42319 |
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