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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 1915 | . 2 ⊢ Ⅎ𝑧𝜑 | |
2 | nfv 1915 | . 2 ⊢ Ⅎ𝑤𝜑 | |
3 | nfv 1915 | . 2 ⊢ Ⅎ𝑥𝜓 | |
4 | nfv 1915 | . 2 ⊢ Ⅎ𝑦𝜓 | |
5 | cbvopabv.1 | . 2 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓)) | |
6 | 1, 2, 3, 4, 5 | cbvopab 5139 | 1 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑧, 𝑤〉 ∣ 𝜓} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 {copab 5130 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-opab 5131 |
This theorem is referenced by: cantnf 9158 infxpen 9442 axdc2 9873 fpwwe2cbv 10054 fpwwecbv 10068 sylow1 18730 bcth 23934 vitali 24216 lgsquadlem3 25960 lgsquad 25961 islnopp 26527 ishpg 26547 hpgbr 26548 trgcopy 26592 trgcopyeu 26594 acopyeu 26622 tgasa1 26646 axcontlem1 26752 eulerpartlemgvv 31636 eulerpart 31642 cvmlift2lem13 32564 pellex 39439 aomclem8 39668 sprsymrelf 43664 |
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