Theorem bj-rcleq 37464

Description: Relative version of dfcleq 2754. (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 2923	. 2 ⊢ Ⅎ𝑥𝐴
2		nfcv 2923	. 2 ⊢ Ⅎ𝑥𝐵
3		nfcv 2923	. 2 ⊢ Ⅎ𝑥𝑉
4	1, 2, 3	bj-rcleqf 37463	1 ⊢ ((𝑉 ∩ 𝐴) = (𝑉 ∩ 𝐵) ↔ ∀𝑥 ∈ 𝑉 (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))

Syntax hints:   ↔ wb 208   = wceq 1559   ∈ wcel 2141  ∀wral 3075   ∩ cin 3903

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-11 2190  ax-12 2211  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-ex 1799  df-nf 1803  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3076  df-v 3455  df-in 3911

This theorem is referenced by: (None)