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| Description: Equality of a class variable and a class abstraction (inference form). (Contributed by NM, 3-Apr-1996.) (Proof shortened by Wolf Lammen, 15-Nov-2019.) | 
| Ref | Expression | 
|---|---|
| eqabri.1 | ⊢ 𝐴 = {𝑥 ∣ 𝜑} | 
| Ref | Expression | 
|---|---|
| eqabri | ⊢ (𝑥 ∈ 𝐴 ↔ 𝜑) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqabri.1 | . . . 4 ⊢ 𝐴 = {𝑥 ∣ 𝜑} | |
| 2 | 1 | a1i 11 | . . 3 ⊢ (⊤ → 𝐴 = {𝑥 ∣ 𝜑}) | 
| 3 | 2 | eqabrd 2883 | . 2 ⊢ (⊤ → (𝑥 ∈ 𝐴 ↔ 𝜑)) | 
| 4 | 3 | mptru 1546 | 1 ⊢ (𝑥 ∈ 𝐴 ↔ 𝜑) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 = wceq 1539 ⊤wtru 1540 ∈ wcel 2107 {cab 2713 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-12 2176 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 | 
| This theorem is referenced by: eqabcri 2885 rabid 3457 csbcow 3913 csbco 3914 csbgfi 3918 csbnestgfw 4421 csbnestgf 4426 ssrelOLD 5792 relopabi 5831 cnv0 6159 funcnv3 6635 opabiota 6990 zfrep6 7980 frrlem2 8313 frrlem3 8314 frrlem4 8315 frrlem8 8319 fprresex 8336 wfrlem2OLD 8350 wfrlem3OLD 8351 wfrlem4OLD 8353 wfrdmclOLD 8358 tfrlem4 8420 tfrlem8 8425 tfrlem9 8426 ixpn0 8971 sbthlem1 9124 dffi3 9472 1idpr 11070 ltexprlem1 11077 ltexprlem2 11078 ltexprlem3 11079 ltexprlem4 11080 ltexprlem6 11082 ltexprlem7 11083 reclem2pr 11089 reclem3pr 11090 reclem4pr 11091 supsrlem 11152 dissnref 23537 dissnlocfin 23538 txbas 23576 xkoccn 23628 xkoptsub 23663 xkoco1cn 23666 xkoco2cn 23667 xkoinjcn 23696 mbfi1fseqlem4 25754 avril1 30483 rnmposs 32685 bnj1436 34854 bnj916 34948 bnj983 34966 bnj1083 34993 bnj1245 35029 bnj1311 35039 bnj1371 35044 bnj1398 35049 setinds 35780 bj-elsngl 36970 bj-projun 36996 bj-projval 36998 f1omptsnlem 37338 icoreresf 37354 finxp0 37393 finxp1o 37394 finxpsuclem 37399 sdclem1 37751 csbcom2fi 38136 rr-grothshortbi 44327 modelaxreplem3 45002 upwordisword 46901 tworepnotupword 46906 | 
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