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Mirrors > Home > MPE Home > Th. List > elab2 | Structured version Visualization version GIF version |
Description: Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.) |
Ref | Expression |
---|---|
elab2.1 | ⊢ 𝐴 ∈ V |
elab2.2 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
elab2.3 | ⊢ 𝐵 = {𝑥 ∣ 𝜑} |
Ref | Expression |
---|---|
elab2 | ⊢ (𝐴 ∈ 𝐵 ↔ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elab2.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | elab2.2 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
3 | elab2.3 | . . 3 ⊢ 𝐵 = {𝑥 ∣ 𝜑} | |
4 | 2, 3 | elab2g 3671 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝐵 ↔ 𝜓)) |
5 | 1, 4 | ax-mp 5 | 1 ⊢ (𝐴 ∈ 𝐵 ↔ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2107 {cab 2710 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-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 |
This theorem is referenced by: opabidw 5525 opabid 5526 oprabidw 7440 oprabid 7441 soseq 8145 wfrlem3OLDa 8311 tfrlem3a 8377 fsetfcdm 8854 cardprclem 9974 iunfictbso 10109 aceq3lem 10115 dfac5lem4 10121 kmlem9 10153 domtriomlem 10437 ltexprlem3 11033 ltexprlem4 11034 reclem2pr 11043 reclem3pr 11044 supsrlem 11106 supaddc 12181 supadd 12182 supmul1 12183 supmullem1 12184 supmullem2 12185 supmul 12186 01sqrexlem6 15194 infcvgaux2i 15804 mertenslem1 15830 mertenslem2 15831 4sqlem12 16889 conjnmzb 19127 sylow3lem2 19496 mdetunilem9 22122 txuni2 23069 xkoopn 23093 met2ndci 24031 2sqlem8 26929 2sqlem11 26932 madef 27351 eulerpartlemt 33370 eulerpartlemr 33373 eulerpartlemn 33380 subfacp1lem3 34173 subfacp1lem5 34175 rdgssun 36259 finxpsuclem 36278 heiborlem1 36679 heiborlem6 36684 heiborlem8 36686 cllem0 42317 fsetsnf 45761 fsetsnfo 45763 cfsetsnfsetf 45768 cfsetsnfsetf1 45769 cfsetsnfsetfo 45770 |
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