Theorem csbprc 4349

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 3738	. . . 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 2802	. 2 ⊢ (¬ 𝐴 ∈ V → {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ ⊥})
5		df-csb 3838	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵}
6		dfnul4 4275	. 2 ⊢ ∅ = {𝑦 ∣ ⊥}
7	4, 5, 6	3eqtr4g 2796	1 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1542  ⊥wfal 1554   ∈ wcel 2114  {cab 2714  Vcvv 3429  [wsbc 3728  ⦋csb 3837  ∅c0 4273

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 2708

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 2715  df-cleq 2728  df-clel 2811  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-nul 4274

This theorem is referenced by:  csb0  4350  sbcel12  4351  sbcne12  4355  sbcel2  4358  csbidm  4373  csbun  4381  csbin  4382  csbdif  4465  csbif  4524  csbuni  4880  sbcbr123  5139  sbcbr  5140  csbexg  5245  csbopab  5510  csbxp  5732  csbres  5947  csbima12  6044  csbrn  6167  csbiota  6491  csbfv12  6885  csbfv  6887  csbriota  7339  csbov123  7411  csbov  7412  csbttc  36691