Theorem bj-reabeq 35265

Description: Relative form of abeq2 2870. (Contributed by BJ, 27-Dec-2023.)

Assertion

Ref	Expression
bj-reabeq	⊢ ((𝑉 ∩ 𝐴) = {𝑥 ∈ 𝑉 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝑉 (𝑥 ∈ 𝐴 ↔ 𝜑))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑉

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem bj-reabeq

Step	Hyp	Ref	Expression
1		dfrab3 4249	. . 3 ⊢ {𝑥 ∈ 𝑉 ∣ 𝜑} = (𝑉 ∩ {𝑥 ∣ 𝜑})
2	1	eqeq2i 2749	. 2 ⊢ ((𝑉 ∩ 𝐴) = {𝑥 ∈ 𝑉 ∣ 𝜑} ↔ (𝑉 ∩ 𝐴) = (𝑉 ∩ {𝑥 ∣ 𝜑}))
3		nfcv 2905	. . 3 ⊢ Ⅎ𝑥𝐴
4		nfab1 2907	. . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑}
5		nfcv 2905	. . 3 ⊢ Ⅎ𝑥𝑉
6	3, 4, 5	bj-rcleqf 35263	. 2 ⊢ ((𝑉 ∩ 𝐴) = (𝑉 ∩ {𝑥 ∣ 𝜑}) ↔ ∀𝑥 ∈ 𝑉 (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝜑}))
7		abid 2717	. . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
8	7	bibi2i 338	. . 3 ⊢ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝜑}) ↔ (𝑥 ∈ 𝐴 ↔ 𝜑))
9	8	ralbii 3093	. 2 ⊢ (∀𝑥 ∈ 𝑉 (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝜑}) ↔ ∀𝑥 ∈ 𝑉 (𝑥 ∈ 𝐴 ↔ 𝜑))
10	2, 6, 9	3bitri 297	1 ⊢ ((𝑉 ∩ 𝐴) = {𝑥 ∈ 𝑉 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝑉 (𝑥 ∈ 𝐴 ↔ 𝜑))

Syntax hints:   ↔ wb 205   = wceq 1539   ∈ wcel 2104  {cab 2713  ∀wral 3062  {crab 3303   ∩ cin 3891

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-tru 1542  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ral 3063  df-rab 3306  df-v 3439  df-in 3899

This theorem is referenced by: (None)