Mathbox for Steven Nguyen |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > rabdif | Structured version Visualization version GIF version |
Description: Move difference in and out of a restricted class abstraction. (Contributed by Steven Nguyen, 6-Jun-2023.) |
Ref | Expression |
---|---|
rabdif | ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∖ 𝐵) = {𝑥 ∈ (𝐴 ∖ 𝐵) ∣ 𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | indif2 4210 | . 2 ⊢ ({𝑥 ∣ 𝜑} ∩ (𝐴 ∖ 𝐵)) = (({𝑥 ∣ 𝜑} ∩ 𝐴) ∖ 𝐵) | |
2 | dfrab2 4250 | . 2 ⊢ {𝑥 ∈ (𝐴 ∖ 𝐵) ∣ 𝜑} = ({𝑥 ∣ 𝜑} ∩ (𝐴 ∖ 𝐵)) | |
3 | dfrab2 4250 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = ({𝑥 ∣ 𝜑} ∩ 𝐴) | |
4 | 3 | difeq1i 4059 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∖ 𝐵) = (({𝑥 ∣ 𝜑} ∩ 𝐴) ∖ 𝐵) |
5 | 1, 2, 4 | 3eqtr4ri 2775 | 1 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∖ 𝐵) = {𝑥 ∈ (𝐴 ∖ 𝐵) ∣ 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 {cab 2713 {crab 3284 ∖ cdif 3889 ∩ cin 3891 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1542 df-ex 1780 df-sb 2066 df-clab 2714 df-cleq 2728 df-clel 2814 df-rab 3287 df-v 3439 df-dif 3895 df-in 3899 |
This theorem is referenced by: prjspeclsp 40488 |
Copyright terms: Public domain | W3C validator |