Theorem bj-inrab3 32117

Description: Generalization of dfrab3ss 3864, which it may shorten. (Contributed by BJ, 21-Apr-2019.) (Revised by OpenAI, 7-Jul-2020.)

Assertion

Ref	Expression
bj-inrab3	⊢ (𝐴 ∩ {𝑥 ∈ 𝐵 ∣ 𝜑}) = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ 𝐵)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem bj-inrab3

Step	Hyp	Ref	Expression
1		dfrab3 3861	. . 3 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} = (𝐵 ∩ {𝑥 ∣ 𝜑})
2	1	ineq2i 3773	. 2 ⊢ (𝐴 ∩ {𝑥 ∈ 𝐵 ∣ 𝜑}) = (𝐴 ∩ (𝐵 ∩ {𝑥 ∣ 𝜑}))
3		incom 3767	. . 3 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ 𝐵) = (𝐵 ∩ {𝑥 ∈ 𝐴 ∣ 𝜑})
4		dfrab3 3861	. . . . 5 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = (𝐴 ∩ {𝑥 ∣ 𝜑})
5	4	ineq2i 3773	. . . 4 ⊢ (𝐵 ∩ {𝑥 ∈ 𝐴 ∣ 𝜑}) = (𝐵 ∩ (𝐴 ∩ {𝑥 ∣ 𝜑}))
6		in12 3786	. . . 4 ⊢ (𝐴 ∩ (𝐵 ∩ {𝑥 ∣ 𝜑})) = (𝐵 ∩ (𝐴 ∩ {𝑥 ∣ 𝜑}))
7	5, 6	eqtr4i 2635	. . 3 ⊢ (𝐵 ∩ {𝑥 ∈ 𝐴 ∣ 𝜑}) = (𝐴 ∩ (𝐵 ∩ {𝑥 ∣ 𝜑}))
8	3, 7	eqtri 2632	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ 𝐵) = (𝐴 ∩ (𝐵 ∩ {𝑥 ∣ 𝜑}))
9	2, 8	eqtr4i 2635	1 ⊢ (𝐴 ∩ {𝑥 ∈ 𝐵 ∣ 𝜑}) = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ 𝐵)

Syntax hints:   = wceq 1475  {cab 2596  {crab 2900   ∩ cin 3539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-in 3547

This theorem is referenced by: (None)