Theorem bj-rcleq 37523

Description: Relative version of dfcleq 2758. (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 2927	. 2 ⊢ Ⅎ𝑥𝐴
2		nfcv 2927	. 2 ⊢ Ⅎ𝑥𝐵
3		nfcv 2927	. 2 ⊢ Ⅎ𝑥𝑉
4	1, 2, 3	bj-rcleqf 37522	1 ⊢ ((𝑉 ∩ 𝐴) = (𝑉 ∩ 𝐵) ↔ ∀𝑥 ∈ 𝑉 (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))

Syntax hints:   ↔ wb 209   = wceq 1563   ∈ wcel 2145  ∀wral 3079   ∩ cin 3906

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-11 2194  ax-12 2215  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1566  df-ex 1803  df-nf 1807  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3080  df-v 3459  df-in 3914

This theorem is referenced by: (None)