Theorem bj-rcleq 36994

Description: Relative version of dfcleq 2733. (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 36993	1 ⊢ ((𝑉 ∩ 𝐴) = (𝑉 ∩ 𝐵) ↔ ∀𝑥 ∈ 𝑉 (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))

Syntax hints:   ↔ wb 206   = wceq 1537   ∈ wcel 2108  ∀wral 3067   ∩ cin 3975

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2158  ax-12 2178  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-v 3490  df-in 3983

This theorem is referenced by: (None)