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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-rcleq | Structured version Visualization version GIF version |
Description: Relative version of dfcleq 2719. (Contributed by BJ, 27-Dec-2023.) |
Ref | Expression |
---|---|
bj-rcleq | ⊢ ((𝑉 ∩ 𝐴) = (𝑉 ∩ 𝐵) ↔ ∀𝑥 ∈ 𝑉 (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2897 | . 2 ⊢ Ⅎ𝑥𝐴 | |
2 | nfcv 2897 | . 2 ⊢ Ⅎ𝑥𝐵 | |
3 | nfcv 2897 | . 2 ⊢ Ⅎ𝑥𝑉 | |
4 | 1, 2, 3 | bj-rcleqf 36412 | 1 ⊢ ((𝑉 ∩ 𝐴) = (𝑉 ∩ 𝐵) ↔ ∀𝑥 ∈ 𝑉 (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1533 ∈ wcel 2098 ∀wral 3055 ∩ cin 3942 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ral 3056 df-rab 3427 df-v 3470 df-in 3950 |
This theorem is referenced by: (None) |
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