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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 form). (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 1906 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | nfv 1906 | . 2 ⊢ Ⅎ𝑦𝜑 | |
3 | nfv 1906 | . 2 ⊢ Ⅎ𝑧𝜑 | |
4 | oprabbidv.1 | . 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
5 | 1, 2, 3, 4 | oprabbid 7208 | 1 ⊢ (𝜑 → {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜒}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 = wceq 1528 {coprab 7146 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-9 2115 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-oprab 7149 |
This theorem is referenced by: oprabbii 7210 mpoeq123dva 7217 mpoeq3dva 7220 resoprab2 7260 erovlem 8382 joinfval 17599 meetfval 17613 odumeet 17738 odujoin 17740 mppsval 32716 csbmpo123 34494 unceq 34750 uncf 34752 unccur 34756 |
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