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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 1915 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | nfv 1915 | . 2 ⊢ Ⅎ𝑦𝜑 | |
3 | nfv 1915 | . 2 ⊢ Ⅎ𝑧𝜑 | |
4 | oprabbidv.1 | . 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
5 | 1, 2, 3, 4 | oprabbid 7198 | 1 ⊢ (𝜑 → {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜒}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 {coprab 7136 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-9 2121 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-oprab 7139 |
This theorem is referenced by: oprabbii 7200 mpoeq123dva 7207 mpoeq3dva 7210 resoprab2 7250 erovlem 8376 joinfval 17603 meetfval 17617 odumeet 17742 odujoin 17744 mppsval 32932 csbmpo123 34748 unceq 35034 uncf 35036 unccur 35040 |
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