Theorem csbprc 4409

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 3798	. . . 4 ⊢ ([𝐴 / 𝑥]𝑦 ∈ 𝐵 → 𝐴 ∈ V)
2		falim 1557	. . . 4 ⊢ (⊥ → 𝐴 ∈ V)
3	1, 2	pm5.21ni 377	. . 3 ⊢ (¬ 𝐴 ∈ V → ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ ⊥))
4	3	abbidv 2808	. 2 ⊢ (¬ 𝐴 ∈ V → {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ ⊥})
5		df-csb 3900	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵}
6		fal 1554	. . . 4 ⊢ ¬ ⊥
7	6	abf 4406	. . 3 ⊢ {𝑦 ∣ ⊥} = ∅
8	7	eqcomi 2746	. 2 ⊢ ∅ = {𝑦 ∣ ⊥}
9	4, 5, 8	3eqtr4g 2802	1 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540  ⊥wfal 1552   ∈ wcel 2108  {cab 2714  Vcvv 3480  [wsbc 3788  ⦋csb 3899  ∅c0 4333

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-nul 4334

This theorem is referenced by:  csb0  4410  sbcel12  4411  sbcne12  4415  sbcel2  4418  csbidm  4433  csbun  4441  csbin  4442  csbdif  4524  csbif  4583  csbuni  4936  sbcbr123  5197  sbcbr  5198  csbexg  5310  csbopab  5560  csbxp  5785  csbres  6000  csbima12  6097  csbrn  6223  csbiota  6554  csbfv12  6954  csbfv  6956  csbriota  7403  csbov123  7475  csbov  7476