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Mirrors > Home > MPE Home > Th. List > cbvopabv | Structured version Visualization version GIF version |
Description: Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 15-Oct-1996.) |
Ref | Expression |
---|---|
cbvopabv.1 | ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvopabv | ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑧, 𝑤〉 ∣ 𝜓} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1906 | . 2 ⊢ Ⅎ𝑧𝜑 | |
2 | nfv 1906 | . 2 ⊢ Ⅎ𝑤𝜑 | |
3 | nfv 1906 | . 2 ⊢ Ⅎ𝑥𝜓 | |
4 | nfv 1906 | . 2 ⊢ Ⅎ𝑦𝜓 | |
5 | cbvopabv.1 | . 2 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓)) | |
6 | 1, 2, 3, 4, 5 | cbvopab 5128 | 1 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑧, 𝑤〉 ∣ 𝜓} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 {copab 5119 |
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-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-opab 5120 |
This theorem is referenced by: cantnf 9144 infxpen 9428 axdc2 9859 fpwwe2cbv 10040 fpwwecbv 10054 sylow1 18657 bcth 23859 vitali 24141 lgsquadlem3 25885 lgsquad 25886 islnopp 26452 ishpg 26472 hpgbr 26473 trgcopy 26517 trgcopyeu 26519 acopyeu 26547 tgasa1 26571 axcontlem1 26677 eulerpartlemgvv 31533 eulerpart 31539 cvmlift2lem13 32459 pellex 39310 aomclem8 39539 sprsymrelf 43534 |
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