Theorem rabdif 4274

Description: Move difference in and out of a restricted class abstraction. (Contributed by Steven Nguyen, 6-Jun-2023.)

Assertion

Ref	Expression
rabdif	⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∖ 𝐵) = {𝑥 ∈ (𝐴 ∖ 𝐵) ∣ 𝜑}

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rabdif

Step	Hyp	Ref	Expression
1		indif2 4234	. 2 ⊢ ({𝑥 ∣ 𝜑} ∩ (𝐴 ∖ 𝐵)) = (({𝑥 ∣ 𝜑} ∩ 𝐴) ∖ 𝐵)
2		dfrab2 4273	. 2 ⊢ {𝑥 ∈ (𝐴 ∖ 𝐵) ∣ 𝜑} = ({𝑥 ∣ 𝜑} ∩ (𝐴 ∖ 𝐵))
3		dfrab2 4273	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = ({𝑥 ∣ 𝜑} ∩ 𝐴)
4	3	difeq1i 4077	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∖ 𝐵) = (({𝑥 ∣ 𝜑} ∩ 𝐴) ∖ 𝐵)
5	1, 2, 4	3eqtr4ri 2799	1 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∖ 𝐵) = {𝑥 ∈ (𝐴 ∖ 𝐵) ∣ 𝜑}

This proof depends on syntax axioms:   = wceq 1570  {cab 2743  {crab 3418   ∖ cdif 3903   ∩ cin 3905

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3419  df-v 3459  df-dif 3909  df-in 3913

This theorem is used by:  prjspeclsp  43377  dfclnbgr4  48622  clnbupgr  48631  dfnbgr5  48649  dfnbgr6  48655