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Mirrors > Home > ILE Home > Th. List > oprabbidv | GIF version |
Description: Equivalent wff's yield equal operation class abstractions (deduction rule). (Contributed by NM, 21-Feb-2004.) |
Ref | Expression |
---|---|
oprabbidv.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
oprabbidv | ⊢ (𝜑 → {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜒}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1421 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | nfv 1421 | . 2 ⊢ Ⅎ𝑦𝜑 | |
3 | nfv 1421 | . 2 ⊢ Ⅎ𝑧𝜑 | |
4 | oprabbidv.1 | . 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
5 | 1, 2, 3, 4 | oprabbid 5558 | 1 ⊢ (𝜑 → {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜒}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 = wceq 1243 {coprab 5513 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-11 1397 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-oprab 5516 |
This theorem is referenced by: oprabbii 5560 mpt2eq123dva 5566 mpt2eq3dva 5569 resoprab2 5598 erovlem 6198 |
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