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Mirrors > Home > ILE Home > Th. List > abbi2i | GIF version |
Description: Equality of a class variable and a class abstraction (inference form). (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
abbiri.1 | ⊢ (𝑥 ∈ 𝐴 ↔ 𝜑) |
Ref | Expression |
---|---|
abbi2i | ⊢ 𝐴 = {𝑥 ∣ 𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abeq2 2226 | . 2 ⊢ (𝐴 = {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜑)) | |
2 | abbiri.1 | . 2 ⊢ (𝑥 ∈ 𝐴 ↔ 𝜑) | |
3 | 1, 2 | mpgbir 1414 | 1 ⊢ 𝐴 = {𝑥 ∣ 𝜑} |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1316 ∈ wcel 1465 {cab 2103 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-11 1469 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 |
This theorem is referenced by: abid2 2238 cbvralcsf 3032 cbvrexcsf 3033 cbvreucsf 3034 cbvrabcsf 3035 symdifxor 3312 dfnul2 3335 dfpr2 3516 dftp2 3542 0iin 3841 pwpwab 3870 epse 4234 fv3 5412 fo1st 6023 fo2nd 6024 xp2 6039 tfrlem3 6176 tfr1onlem3 6203 mapsn 6552 ixpconstg 6569 ixp0x 6588 nnzrab 9046 nn0zrab 9047 |
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