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Mirrors > Home > ILE Home > Th. List > abeq2i | GIF version |
Description: Equality of a class variable and a class abstraction (inference form). (Contributed by NM, 3-Apr-1996.) |
Ref | Expression |
---|---|
abeqi.1 | ⊢ 𝐴 = {𝑥 ∣ 𝜑} |
Ref | Expression |
---|---|
abeq2i | ⊢ (𝑥 ∈ 𝐴 ↔ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abeqi.1 | . . 3 ⊢ 𝐴 = {𝑥 ∣ 𝜑} | |
2 | 1 | eleq2i 2155 | . 2 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝜑}) |
3 | abid 2077 | . 2 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) | |
4 | 2, 3 | bitri 183 | 1 ⊢ (𝑥 ∈ 𝐴 ↔ 𝜑) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1290 ∈ wcel 1439 {cab 2075 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1382 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-ext 2071 |
This theorem depends on definitions: df-bi 116 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 |
This theorem is referenced by: rabid 2543 vex 2623 csbco 2943 csbnestgf 2981 ifmdc 3432 pwss 3449 snsspw 3614 iunpw 4315 ordon 4316 funcnv3 5089 tfrlem4 6092 tfrlem8 6097 tfrlem9 6098 tfrlemibxssdm 6106 tfr1onlembxssdm 6122 tfrcllembxssdm 6135 ixpm 6501 mapsnen 6582 sbthlem1 6720 1idprl 7203 1idpru 7204 recexprlem1ssl 7246 recexprlem1ssu 7247 recexprlemss1l 7248 recexprlemss1u 7249 |
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