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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 3493 | . 2 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} → 𝐴 ∈ V) | |
2 | elex 3493 | . . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) | |
3 | 2 | adantr 482 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → 𝐴 ∈ V) |
4 | df-rab 3434 | . . . 4 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} | |
5 | 4 | eleq2i 2826 | . . 3 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ 𝐴 ∈ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)}) |
6 | elrabf.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
7 | elrabf.2 | . . . . . 6 ⊢ Ⅎ𝑥𝐵 | |
8 | 6, 7 | nfel 2918 | . . . . 5 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵 |
9 | elrabf.3 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
10 | 8, 9 | nfan 1903 | . . . 4 ⊢ Ⅎ𝑥(𝐴 ∈ 𝐵 ∧ 𝜓) |
11 | eleq1 2822 | . . . . 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 3665 | . . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} ↔ (𝐴 ∈ 𝐵 ∧ 𝜓))) |
15 | 5, 14 | bitrid 283 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ (𝐴 ∈ 𝐵 ∧ 𝜓))) |
16 | 1, 3, 15 | pm5.21nii 380 | 1 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ (𝐴 ∈ 𝐵 ∧ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 Ⅎwnf 1786 ∈ wcel 2107 {cab 2710 Ⅎwnfc 2884 {crab 3433 Vcvv 3475 |
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 2704 |
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-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-rab 3434 df-v 3477 |
This theorem is referenced by: rabtru 3681 invdisjrabw 5134 invdisjrab 5135 rabxfrd 5416 f1ossf1o 7126 onminsb 7782 nnawordex 8637 tskwe 9945 rabssnn0fi 13951 iundisj 25065 sltval2 27159 iundisjf 31851 iundisjfi 32038 bnj1388 34075 phpreu 36520 poimirlem26 36562 sticksstones1 41010 rfcnpre3 43765 rfcnpre4 43766 uzwo4 43788 disjinfi 43939 allbutfiinf 44178 fsumiunss 44339 fnlimfvre 44438 stoweidlem26 44790 stoweidlem27 44791 stoweidlem31 44795 stoweidlem34 44798 stoweidlem51 44815 stoweidlem52 44816 stoweidlem59 44823 fourierdlem20 44891 fourierdlem79 44949 pimdecfgtioc 45479 smfpimcclem 45571 prmdvdsfmtnof1lem1 46300 |
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