Theorem csbprc 4366

Description: The proper substitution of a proper class for a set into a class results in the empty set. (Contributed by NM, 17-Aug-2018.) (Proof shortened by JJ, 27-Aug-2021.)

Assertion

Ref	Expression
csbprc	⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅)

Proof of Theorem csbprc

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbcex 3757	. . . 4 ⊢ ([𝐴 / 𝑥]𝑦 ∈ 𝐵 → 𝐴 ∈ V)
2		falim 1580	. . . 4 ⊢ (⊥ → 𝐴 ∈ V)
3	1, 2	pm5.21ni 380	. . 3 ⊢ (¬ 𝐴 ∈ V → ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ ⊥))
4	3	abbidv 2831	. 2 ⊢ (¬ 𝐴 ∈ V → {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ ⊥})
5		df-csb 3856	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵}
6		dfnul4 4290	. 2 ⊢ ∅ = {𝑦 ∣ ⊥}
7	4, 5, 6	3eqtr4g 2825	1 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1563  ⊥wfal 1575   ∈ wcel 2145  {cab 2743  Vcvv 3457  [wsbc 3747  ⦋csb 3855  ∅c0 4288

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-nul 4289

This theorem is referenced by:  csb0  4367  sbcel12  4368  sbcne12  4372  sbcel2  4375  csbidm  4390  csbun  4398  csbin  4399  csbdif  4482  csbif  4541  csbuni  4899  sbcbr123  5159  sbcbr  5160  csbexg  5265  csbopab  5531  csbxp  5753  csbcnv  5863  csbres  5972  csbima12  6072  csbrn  6194  csbiota  6518  csbfv12  6916  csbfv  6918  csbriota  7372  csbov123  7444  csbov  7445  csbttc  36882