![]() |
Mathbox for Peter Mazsa |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > abeqin | Structured version Visualization version GIF version |
Description: Intersection with class abstraction. (Contributed by Peter Mazsa, 21-Jul-2021.) |
Ref | Expression |
---|---|
abeqin.1 | ⊢ 𝐴 = (𝐵 ∩ 𝐶) |
abeqin.2 | ⊢ 𝐵 = {𝑥 ∣ 𝜑} |
Ref | Expression |
---|---|
abeqin | ⊢ 𝐴 = {𝑥 ∈ 𝐶 ∣ 𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abeqin.2 | . . 3 ⊢ 𝐵 = {𝑥 ∣ 𝜑} | |
2 | 1 | ineq1i 4208 | . 2 ⊢ (𝐵 ∩ 𝐶) = ({𝑥 ∣ 𝜑} ∩ 𝐶) |
3 | abeqin.1 | . 2 ⊢ 𝐴 = (𝐵 ∩ 𝐶) | |
4 | dfrab2 4311 | . 2 ⊢ {𝑥 ∈ 𝐶 ∣ 𝜑} = ({𝑥 ∣ 𝜑} ∩ 𝐶) | |
5 | 2, 3, 4 | 3eqtr4i 2766 | 1 ⊢ 𝐴 = {𝑥 ∈ 𝐶 ∣ 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 {cab 2705 {crab 3429 ∩ 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-in 3954 |
This theorem is referenced by: abeqinbi 37725 dfcnvrefrels3 38001 dffunsALTV 38155 dfdisjs 38180 |
Copyright terms: Public domain | W3C validator |