Theorem bj-inrab3 37073

Description: Generalization of dfrab3ss 4273, 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 4269	. . 3 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} = (𝐵 ∩ {𝑥 ∣ 𝜑})
2	1	ineq2i 4167	. 2 ⊢ (𝐴 ∩ {𝑥 ∈ 𝐵 ∣ 𝜑}) = (𝐴 ∩ (𝐵 ∩ {𝑥 ∣ 𝜑}))
3		dfrab3 4269	. . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = (𝐴 ∩ {𝑥 ∣ 𝜑})
4	3	ineq2i 4167	. . 3 ⊢ (𝐵 ∩ {𝑥 ∈ 𝐴 ∣ 𝜑}) = (𝐵 ∩ (𝐴 ∩ {𝑥 ∣ 𝜑}))
5		incom 4159	. . 3 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ 𝐵) = (𝐵 ∩ {𝑥 ∈ 𝐴 ∣ 𝜑})
6		in12 4179	. . 3 ⊢ (𝐴 ∩ (𝐵 ∩ {𝑥 ∣ 𝜑})) = (𝐵 ∩ (𝐴 ∩ {𝑥 ∣ 𝜑}))
7	4, 5, 6	3eqtr4i 2767	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ 𝐵) = (𝐴 ∩ (𝐵 ∩ {𝑥 ∣ 𝜑}))
8	2, 7	eqtr4i 2760	1 ⊢ (𝐴 ∩ {𝑥 ∈ 𝐵 ∣ 𝜑}) = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ 𝐵)

Syntax hints:   = wceq 1541  {cab 2712  {crab 3397   ∩ cin 3898

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-rab 3398  df-v 3440  df-in 3906

This theorem is referenced by: (None)