Theorem bj-reabeq 37691

Description: Relative form of eqabb 2902. (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 2776	. 2 ⊢ ((𝑉 ∩ 𝐴) = {𝑥 ∈ 𝑉 ∣ 𝜑} ↔ (𝑉 ∩ 𝐴) = (𝑉 ∩ {𝑥 ∣ 𝜑}))
3		nfcv 2925	. . 3 ⊢ Ⅎ𝑥𝐴
4		nfab1 2927	. . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑}
5		nfcv 2925	. . 3 ⊢ Ⅎ𝑥𝑉
6	3, 4, 5	bj-rcleqf 37689	. 2 ⊢ ((𝑉 ∩ 𝐴) = (𝑉 ∩ {𝑥 ∣ 𝜑}) ↔ ∀𝑥 ∈ 𝑉 (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝜑}))
7		abid 2745	. . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
8	7	bibi2i 340	. . 3 ⊢ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝜑}) ↔ (𝑥 ∈ 𝐴 ↔ 𝜑))
9	8	ralbii 3111	. 2 ⊢ (∀𝑥 ∈ 𝑉 (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝜑}) ↔ ∀𝑥 ∈ 𝑉 (𝑥 ∈ 𝐴 ↔ 𝜑))
10	2, 6, 9	3bitri 300	1 ⊢ ((𝑉 ∩ 𝐴) = {𝑥 ∈ 𝑉 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝑉 (𝑥 ∈ 𝐴 ↔ 𝜑))

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570   ∈ wcel 2143  {cab 2741  ∀wral 3079  {crab 3416   ∩ cin 3904

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rab 3417  df-v 3457  df-in 3912

This theorem is used by: (None)