Theorem bj-reabeq 36415

Description: Relative form of eqabb 2867. (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 4304	. . 3 ⊢ {𝑥 ∈ 𝑉 ∣ 𝜑} = (𝑉 ∩ {𝑥 ∣ 𝜑})
2	1	eqeq2i 2739	. 2 ⊢ ((𝑉 ∩ 𝐴) = {𝑥 ∈ 𝑉 ∣ 𝜑} ↔ (𝑉 ∩ 𝐴) = (𝑉 ∩ {𝑥 ∣ 𝜑}))
3		nfcv 2897	. . 3 ⊢ Ⅎ𝑥𝐴
4		nfab1 2899	. . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑}
5		nfcv 2897	. . 3 ⊢ Ⅎ𝑥𝑉
6	3, 4, 5	bj-rcleqf 36413	. 2 ⊢ ((𝑉 ∩ 𝐴) = (𝑉 ∩ {𝑥 ∣ 𝜑}) ↔ ∀𝑥 ∈ 𝑉 (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝜑}))
7		abid 2707	. . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
8	7	bibi2i 337	. . 3 ⊢ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝜑}) ↔ (𝑥 ∈ 𝐴 ↔ 𝜑))
9	8	ralbii 3087	. 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 1533   ∈ wcel 2098  {cab 2703  ∀wral 3055  {crab 3426   ∩ cin 3942

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rab 3427  df-v 3470  df-in 3950

This theorem is referenced by: (None)