Theorem csbprc 4363

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 3752	. . . 4 ⊢ ([𝐴 / 𝑥]𝑦 ∈ 𝐵 → 𝐴 ∈ V)
2		falim 1559	. . . 4 ⊢ (⊥ → 𝐴 ∈ V)
3	1, 2	pm5.21ni 377	. . 3 ⊢ (¬ 𝐴 ∈ V → ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ ⊥))
4	3	abbidv 2803	. 2 ⊢ (¬ 𝐴 ∈ V → {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ ⊥})
5		df-csb 3852	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵}
6		fal 1556	. . . 4 ⊢ ¬ ⊥
7	6	abf 4360	. . 3 ⊢ {𝑦 ∣ ⊥} = ∅
8	7	eqcomi 2746	. 2 ⊢ ∅ = {𝑦 ∣ ⊥}
9	4, 5, 8	3eqtr4g 2797	1 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1542  ⊥wfal 1554   ∈ wcel 2114  {cab 2715  Vcvv 3442  [wsbc 3742  ⦋csb 3851  ∅c0 4287

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-nul 4288

This theorem is referenced by:  csb0  4364  sbcel12  4365  sbcne12  4369  sbcel2  4372  csbidm  4387  csbun  4395  csbin  4396  csbdif  4480  csbif  4539  csbuni  4895  sbcbr123  5154  sbcbr  5155  csbexg  5257  csbopab  5511  csbxp  5733  csbres  5949  csbima12  6046  csbrn  6169  csbiota  6493  csbfv12  6887  csbfv  6889  csbriota  7340  csbov123  7412  csbov  7413