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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 2170 | . 2 ⊢ 𝑧 = 𝑧 | |
2 | opabbii.1 | . . . 4 ⊢ (𝜑 ↔ 𝜓) | |
3 | 2 | a1i 9 | . . 3 ⊢ (𝑧 = 𝑧 → (𝜑 ↔ 𝜓)) |
4 | 3 | opabbidv 4053 | . 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 104 = wceq 1348 {copab 4047 |
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 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-11 1499 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-opab 4049 |
This theorem is referenced by: mptv 4084 fconstmpt 4656 xpundi 4665 xpundir 4666 inxp 4743 cnvco 4794 resopab 4933 opabresid 4942 cnvi 5013 cnvun 5014 cnvin 5016 cnvxp 5027 cnvcnv3 5058 coundi 5110 coundir 5111 mptun 5327 fvopab6 5590 cbvoprab1 5923 cbvoprab12 5925 dmoprabss 5933 mpomptx 5942 resoprab 5947 ov6g 5988 dfoprab3s 6167 dfoprab3 6168 dfoprab4 6169 mapsncnv 6670 xpcomco 6801 dmaddpq 7330 dmmulpq 7331 recmulnqg 7342 enq0enq 7382 ltrelxr 7969 ltxr 9721 shftidt2 10785 lmfval 12947 lmbr 12968 cnmptid 13036 |
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