Theorem bj-rcleq 35195

Description: Relative version of dfcleq 2732. (Contributed by BJ, 27-Dec-2023.)

Assertion

Ref	Expression
bj-rcleq	⊢ ((𝑉 ∩ 𝐴) = (𝑉 ∩ 𝐵) ↔ ∀𝑥 ∈ 𝑉 (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑉

Proof of Theorem bj-rcleq

Step	Hyp	Ref	Expression
1		nfcv 2908	. 2 ⊢ Ⅎ𝑥𝐴
2		nfcv 2908	. 2 ⊢ Ⅎ𝑥𝐵
3		nfcv 2908	. 2 ⊢ Ⅎ𝑥𝑉
4	1, 2, 3	bj-rcleqf 35194	1 ⊢ ((𝑉 ∩ 𝐴) = (𝑉 ∩ 𝐵) ↔ ∀𝑥 ∈ 𝑉 (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))

Syntax hints:   ↔ wb 205   = wceq 1541   ∈ wcel 2109  ∀wral 3065   ∩ cin 3890

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1544  df-ex 1786  df-nf 1790  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ral 3070  df-rab 3074  df-v 3432  df-in 3898

This theorem is referenced by: (None)