Theorem csbprc 4321

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 3704	. . . 4 ⊢ ([𝐴 / 𝑥]𝑦 ∈ 𝐵 → 𝐴 ∈ V)
2		falim 1560	. . . 4 ⊢ (⊥ → 𝐴 ∈ V)
3	1, 2	pm5.21ni 382	. . 3 ⊢ (¬ 𝐴 ∈ V → ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ ⊥))
4	3	abbidv 2807	. 2 ⊢ (¬ 𝐴 ∈ V → {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ ⊥})
5		df-csb 3812	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵}
6		fal 1557	. . . 4 ⊢ ¬ ⊥
7	6	abf 4317	. . 3 ⊢ {𝑦 ∣ ⊥} = ∅
8	7	eqcomi 2746	. 2 ⊢ ∅ = {𝑦 ∣ ⊥}
9	4, 5, 8	3eqtr4g 2803	1 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1543  ⊥wfal 1555   ∈ wcel 2110  {cab 2714  Vcvv 3408  [wsbc 3694  ⦋csb 3811  ∅c0 4237

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-nul 4238

This theorem is referenced by:  csb0  4322  sbcel12  4323  sbcne12  4327  sbcel2  4330  csbidm  4345  csbun  4353  csbin  4354  csbif  4496  csbuni  4850  sbcbr123  5107  sbcbr  5108  csbexg  5203  csbopab  5436  csbxp  5647  csbres  5854  csbima12  5947  csbrn  6066  csbiota  6373  csbfv12  6760  csbfv  6762  csbriota  7186  csbov123  7255  csbov  7256  csbdif  35233