Theorem bj-rcleq 36986

Description: Relative version of dfcleq 2727. (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 2897	. 2 ⊢ Ⅎ𝑥𝐴
2		nfcv 2897	. 2 ⊢ Ⅎ𝑥𝐵
3		nfcv 2897	. 2 ⊢ Ⅎ𝑥𝑉
4	1, 2, 3	bj-rcleqf 36985	1 ⊢ ((𝑉 ∩ 𝐴) = (𝑉 ∩ 𝐵) ↔ ∀𝑥 ∈ 𝑉 (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))

Syntax hints:   ↔ wb 206   = wceq 1539   ∈ wcel 2107  ∀wral 3050   ∩ cin 3930

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-11 2156  ax-12 2176  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ral 3051  df-v 3465  df-in 3938

This theorem is referenced by: (None)