Theorem csb0 4381

Description: The proper substitution of a class into the empty set is the empty set. (Contributed by NM, 18-Aug-2018.)

Assertion

Ref	Expression
csb0	⊢ ⦋𝐴 / 𝑥⦌∅ = ∅

Proof of Theorem csb0

Step	Hyp	Ref	Expression
1		csbconstg 3880	. 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌∅ = ∅)
2		csbprc 4380	. 2 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌∅ = ∅)
3	1, 2	pm2.61i 184	1 ⊢ ⦋𝐴 / 𝑥⦌∅ = ∅

Syntax hints:   = wceq 1567   ∈ wcel 2149  Vcvv 3463  ⦋csb 3861  ∅c0 4294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-nul 4295

This theorem is referenced by:  disjdsct  32989  onfrALTlem5  45143  onfrALTlem4  45144  onfrALTlem5VD  45485  onfrALTlem4VD  45486