Theorem csb0 4378

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 3875	. 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌∅ = ∅)
2		csbprc 4377	. 2 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌∅ = ∅)
3	1, 2	pm2.61i 184	1 ⊢ ⦋𝐴 / 𝑥⦌∅ = ∅

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2146  Vcvv 3458  ⦋csb 3856  ∅c0 4289

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-nul 4290

This theorem is used by:  disjdsct  33085  onfrALTlem5  45292  onfrALTlem4  45293  onfrALTlem5VD  45634  onfrALTlem4VD  45635