Theorem bj-reabeq 37203

Description: Relative form of eqabb 2876. (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 4272	. . 3 ⊢ {𝑥 ∈ 𝑉 ∣ 𝜑} = (𝑉 ∩ {𝑥 ∣ 𝜑})
2	1	eqeq2i 2750	. 2 ⊢ ((𝑉 ∩ 𝐴) = {𝑥 ∈ 𝑉 ∣ 𝜑} ↔ (𝑉 ∩ 𝐴) = (𝑉 ∩ {𝑥 ∣ 𝜑}))
3		nfcv 2899	. . 3 ⊢ Ⅎ𝑥𝐴
4		nfab1 2901	. . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑}
5		nfcv 2899	. . 3 ⊢ Ⅎ𝑥𝑉
6	3, 4, 5	bj-rcleqf 37201	. 2 ⊢ ((𝑉 ∩ 𝐴) = (𝑉 ∩ {𝑥 ∣ 𝜑}) ↔ ∀𝑥 ∈ 𝑉 (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝜑}))
7		abid 2719	. . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
8	7	bibi2i 337	. . 3 ⊢ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝜑}) ↔ (𝑥 ∈ 𝐴 ↔ 𝜑))
9	8	ralbii 3083	. 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 206   = wceq 1542   ∈ wcel 2114  {cab 2715  ∀wral 3052  {crab 3400   ∩ cin 3901

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rab 3401  df-v 3443  df-in 3909

This theorem is referenced by: (None)