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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 1830 | . 2 ⊢ Ⅎ𝑧𝜑 | |
2 | nfv 1830 | . 2 ⊢ Ⅎ𝑤𝜑 | |
3 | nfv 1830 | . 2 ⊢ Ⅎ𝑥𝜓 | |
4 | nfv 1830 | . 2 ⊢ Ⅎ𝑦𝜓 | |
5 | cbvopabv.1 | . 2 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓)) | |
6 | 1, 2, 3, 4, 5 | cbvopab 4653 | 1 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑧, 𝑤〉 ∣ 𝜓} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 {copab 4642 |
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-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-opab 4644 |
This theorem is referenced by: cantnf 8473 infxpen 8720 axdc2 9154 fpwwe2cbv 9331 fpwwecbv 9345 sylow1 17841 bcth 22934 vitali 23188 lgsquadlem3 24907 lgsquad 24908 islnopp 25431 ishpg 25451 hpgbr 25452 trgcopy 25496 trgcopyeu 25498 acopyeu 25525 tgasa1 25539 axcontlem1 25644 eulerpartlemgvv 29765 eulerpart 29771 cvmlift2lem13 30551 pellex 36417 aomclem8 36649 |
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