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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 3512 | . 2 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} → 𝐴 ∈ V) | |
2 | elex 3512 | . . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) | |
3 | 2 | adantr 483 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → 𝐴 ∈ V) |
4 | df-rab 3147 | . . . 4 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} | |
5 | 4 | eleq2i 2904 | . . 3 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ 𝐴 ∈ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)}) |
6 | elrabf.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
7 | elrabf.2 | . . . . . 6 ⊢ Ⅎ𝑥𝐵 | |
8 | 6, 7 | nfel 2992 | . . . . 5 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵 |
9 | elrabf.3 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
10 | 8, 9 | nfan 1896 | . . . 4 ⊢ Ⅎ𝑥(𝐴 ∈ 𝐵 ∧ 𝜓) |
11 | eleq1 2900 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) | |
12 | elrabf.4 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
13 | 11, 12 | anbi12d 632 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 ∧ 𝜑) ↔ (𝐴 ∈ 𝐵 ∧ 𝜓))) |
14 | 6, 10, 13 | elabgf 3663 | . . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} ↔ (𝐴 ∈ 𝐵 ∧ 𝜓))) |
15 | 5, 14 | syl5bb 285 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ (𝐴 ∈ 𝐵 ∧ 𝜓))) |
16 | 1, 3, 15 | pm5.21nii 382 | 1 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ (𝐴 ∈ 𝐵 ∧ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 Ⅎwnf 1780 ∈ wcel 2110 {cab 2799 Ⅎwnfc 2961 {crab 3142 Vcvv 3494 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 |
This theorem is referenced by: rabtru 3676 invdisjrabw 5050 invdisjrab 5051 rabxfrd 5317 f1ossf1o 6889 onminsb 7513 nnawordex 8262 tskwe 9378 rabssnn0fi 13353 iundisj 24148 iundisjf 30338 iundisjfi 30518 bnj1388 32305 sltval2 33163 phpreu 34875 poimirlem26 34917 rfcnpre3 41288 rfcnpre4 41289 uzwo4 41313 disjinfi 41452 allbutfiinf 41692 fsumiunss 41854 fnlimfvre 41953 stoweidlem26 42310 stoweidlem27 42311 stoweidlem31 42315 stoweidlem34 42318 stoweidlem51 42335 stoweidlem52 42336 stoweidlem59 42343 fourierdlem20 42411 fourierdlem79 42469 pimdecfgtioc 42992 smfpimcclem 43080 prmdvdsfmtnof1lem1 43745 |
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