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Mirrors > Home > ILE Home > Th. List > opabbii | GIF version |
Description: Equivalent wff's yield equal class abstractions. (Contributed by NM, 15-May-1995.) |
Ref | Expression |
---|---|
opabbii.1 | ⊢ (𝜑 ↔ 𝜓) |
Ref | Expression |
---|---|
opabbii | ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑥, 𝑦〉 ∣ 𝜓} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2189 | . 2 ⊢ 𝑧 = 𝑧 | |
2 | opabbii.1 | . . . 4 ⊢ (𝜑 ↔ 𝜓) | |
3 | 2 | a1i 9 | . . 3 ⊢ (𝑧 = 𝑧 → (𝜑 ↔ 𝜓)) |
4 | 3 | opabbidv 4084 | . 2 ⊢ (𝑧 = 𝑧 → {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑥, 𝑦〉 ∣ 𝜓}) |
5 | 1, 4 | ax-mp 5 | 1 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑥, 𝑦〉 ∣ 𝜓} |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 = wceq 1364 {copab 4078 |
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-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-opab 4080 |
This theorem is referenced by: mptv 4115 fconstmpt 4688 xpundi 4697 xpundir 4698 inxp 4776 cnvco 4827 resopab 4966 opabresid 4975 cnvi 5048 cnvun 5049 cnvin 5051 cnvxp 5062 cnvcnv3 5093 coundi 5145 coundir 5146 mptun 5362 fvopab6 5628 cbvoprab1 5963 cbvoprab12 5965 dmoprabss 5973 mpomptx 5982 resoprab 5987 ov6g 6029 dfoprab3s 6209 dfoprab3 6210 dfoprab4 6211 mapsncnv 6713 xpcomco 6844 dmaddpq 7396 dmmulpq 7397 recmulnqg 7408 enq0enq 7448 ltrelxr 8036 ltxr 9793 shftidt2 10859 prdsex 12740 releqgg 13125 eqgex 13126 dvdsrzring 13863 lmfval 14089 lmbr 14110 cnmptid 14178 |
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