Theorem csbprc 4362

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 3751	. . . 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 3851	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵}
6		fal 1556	. . . 4 ⊢ ¬ ⊥
7	6	abf 4359	. . 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 3441  [wsbc 3741  ⦋csb 3850  ∅c0 4286

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 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-nul 4287

This theorem is referenced by:  csb0  4363  sbcel12  4364  sbcne12  4368  sbcel2  4371  csbidm  4386  csbun  4394  csbin  4395  csbdif  4479  csbif  4538  csbuni  4894  sbcbr123  5153  sbcbr  5154  csbexg  5256  csbopab  5504  csbxp  5726  csbres  5942  csbima12  6039  csbrn  6162  csbiota  6486  csbfv12  6880  csbfv  6882  csbriota  7332  csbov123  7404  csbov  7405