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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 2298 | . 2 ⊢ (𝐴 = {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜑)) | |
2 | abbiri.1 | . 2 ⊢ (𝑥 ∈ 𝐴 ↔ 𝜑) | |
3 | 1, 2 | mpgbir 1464 | 1 ⊢ 𝐴 = {𝑥 ∣ 𝜑} |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 = wceq 1364 ∈ wcel 2160 {cab 2175 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-11 1517 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 |
This theorem is referenced by: abid2 2310 cbvralcsf 3134 cbvrexcsf 3135 cbvreucsf 3136 cbvrabcsf 3137 symdifxor 3416 dfnul2 3439 dfpr2 3626 dftp2 3656 0iin 3960 pwpwab 3989 epse 4360 fv3 5557 fo1st 6183 fo2nd 6184 xp2 6199 tfrlem3 6337 tfr1onlem3 6364 mapsn 6717 ixpconstg 6734 ixp0x 6753 nnzrab 9308 nn0zrab 9309 |
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