![]() |
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 4271 | . 2 ⊢ ({𝑥 ∣ 𝜑} ∩ (𝐴 ∖ 𝐵)) = (({𝑥 ∣ 𝜑} ∩ 𝐴) ∖ 𝐵) | |
2 | dfrab2 4311 | . 2 ⊢ {𝑥 ∈ (𝐴 ∖ 𝐵) ∣ 𝜑} = ({𝑥 ∣ 𝜑} ∩ (𝐴 ∖ 𝐵)) | |
3 | dfrab2 4311 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = ({𝑥 ∣ 𝜑} ∩ 𝐴) | |
4 | 3 | difeq1i 4116 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∖ 𝐵) = (({𝑥 ∣ 𝜑} ∩ 𝐴) ∖ 𝐵) |
5 | 1, 2, 4 | 3eqtr4ri 2767 | 1 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∖ 𝐵) = {𝑥 ∈ (𝐴 ∖ 𝐵) ∣ 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 {cab 2705 {crab 3429 ∖ cdif 3944 ∩ cin 3946 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-rab 3430 df-v 3473 df-dif 3950 df-in 3954 |
This theorem is referenced by: prjspeclsp 42036 |
Copyright terms: Public domain | W3C validator |