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Mirrors > Home > ILE Home > Th. List > opabbidv | GIF version |
Description: Equivalent wff's yield equal ordered-pair class abstractions (deduction rule). (Contributed by NM, 15-May-1995.) |
Ref | Expression |
---|---|
opabbidv.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
opabbidv | ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ 𝜓} = {〈𝑥, 𝑦〉 ∣ 𝜒}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1421 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | nfv 1421 | . 2 ⊢ Ⅎ𝑦𝜑 | |
3 | opabbidv.1 | . 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
4 | 1, 2, 3 | opabbid 3822 | 1 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ 𝜓} = {〈𝑥, 𝑦〉 ∣ 𝜒}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 = wceq 1243 {copab 3817 |
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-opab 3819 |
This theorem is referenced by: opabbii 3824 csbopabg 3835 xpeq1 4359 xpeq2 4360 opabbi2dv 4485 csbcnvg 4519 resopab2 4655 cores 4824 xpcom 4864 dffn5im 5219 f1oiso2 5466 f1ocnvd 5702 ofreq 5715 sprmpt2 5857 shftfvalg 9419 shftfval 9422 2shfti 9432 |
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