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Mirrors > Home > ILE Home > Th. List > oprabbii | GIF version |
Description: Equivalent wff's yield equal operation class abstractions. (Contributed by NM, 28-May-1995.) (Revised by David Abernethy, 19-Jun-2012.) |
Ref | Expression |
---|---|
oprabbii.1 | ⊢ (𝜑 ↔ 𝜓) |
Ref | Expression |
---|---|
oprabbii | ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2113 | . 2 ⊢ 𝑤 = 𝑤 | |
2 | oprabbii.1 | . . . 4 ⊢ (𝜑 ↔ 𝜓) | |
3 | 2 | a1i 9 | . . 3 ⊢ (𝑤 = 𝑤 → (𝜑 ↔ 𝜓)) |
4 | 3 | oprabbidv 5777 | . 2 ⊢ (𝑤 = 𝑤 → {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓}) |
5 | 1, 4 | ax-mp 7 | 1 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1312 {coprab 5727 |
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 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-11 1465 ax-4 1468 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 |
This theorem depends on definitions: df-bi 116 df-tru 1315 df-nf 1418 df-sb 1717 df-clab 2100 df-cleq 2106 df-oprab 5730 |
This theorem is referenced by: oprab4 5794 mpov 5813 dfxp3 6044 tposmpo 6130 oviec 6487 dfplpq2 7104 dfmpq2 7105 dfmq0qs 7179 dfplq0qs 7180 addsrpr 7482 mulsrpr 7483 addcnsr 7563 mulcnsr 7564 addvalex 7573 |
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