Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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.) Reduce axiom usage. (Revised by Gino Giotto, 15-Oct-2024.) |
Ref | Expression |
---|---|
cbvopabv.1 | ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvopabv | ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑧, 𝑤〉 ∣ 𝜓} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opeq12 4772 | . . . . . 6 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 〈𝑥, 𝑦〉 = 〈𝑧, 𝑤〉) | |
2 | 1 | eqeq2d 2747 | . . . . 5 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑣 = 〈𝑥, 𝑦〉 ↔ 𝑣 = 〈𝑧, 𝑤〉)) |
3 | cbvopabv.1 | . . . . 5 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓)) | |
4 | 2, 3 | anbi12d 634 | . . . 4 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ (𝑣 = 〈𝑧, 𝑤〉 ∧ 𝜓))) |
5 | 4 | cbvex2vw 2051 | . . 3 ⊢ (∃𝑥∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑧∃𝑤(𝑣 = 〈𝑧, 𝑤〉 ∧ 𝜓)) |
6 | 5 | abbii 2801 | . 2 ⊢ {𝑣 ∣ ∃𝑥∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑)} = {𝑣 ∣ ∃𝑧∃𝑤(𝑣 = 〈𝑧, 𝑤〉 ∧ 𝜓)} |
7 | df-opab 5102 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑣 ∣ ∃𝑥∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
8 | df-opab 5102 | . 2 ⊢ {〈𝑧, 𝑤〉 ∣ 𝜓} = {𝑣 ∣ ∃𝑧∃𝑤(𝑣 = 〈𝑧, 𝑤〉 ∧ 𝜓)} | |
9 | 6, 7, 8 | 3eqtr4i 2769 | 1 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑧, 𝑤〉 ∣ 𝜓} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∃wex 1787 {cab 2714 〈cop 4533 {copab 5101 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-opab 5102 |
This theorem is referenced by: cantnf 9286 infxpen 9593 axdc2 10028 fpwwe2cbv 10209 fpwwecbv 10223 sylow1 18946 bcth 24180 vitali 24464 lgsquadlem3 26217 lgsquad 26218 islnopp 26784 ishpg 26804 hpgbr 26805 trgcopy 26849 trgcopyeu 26851 acopyeu 26879 tgasa1 26903 axcontlem1 27009 eulerpartlemgvv 32009 eulerpart 32015 cvmlift2lem13 32944 pellex 40301 aomclem8 40530 sprsymrelf 44563 |
Copyright terms: Public domain | W3C validator |