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Mirrors > Home > MPE Home > Th. List > oprabbidv | Structured version Visualization version 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 1830 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | nfv 1830 | . 2 ⊢ Ⅎ𝑦𝜑 | |
3 | nfv 1830 | . 2 ⊢ Ⅎ𝑧𝜑 | |
4 | oprabbidv.1 | . 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
5 | 1, 2, 3, 4 | oprabbid 6606 | 1 ⊢ (𝜑 → {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜒}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 {coprab 6550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-oprab 6553 |
This theorem is referenced by: oprabbii 6608 mpt2eq123dva 6614 mpt2eq3dva 6617 resoprab2 6655 erovlem 7730 joinfval 16824 meetfval 16838 odumeet 16963 odujoin 16965 mppsval 30723 csbmpt22g 32353 unceq 32556 uncf 32558 unccur 32562 |
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