Theorem csb0 4371

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 3869	. 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌∅ = ∅)
2		csbprc 4370	. 2 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌∅ = ∅)
3	1, 2	pm2.61i 184	1 ⊢ ⦋𝐴 / 𝑥⦌∅ = ∅

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2145  Vcvv 3453  ⦋csb 3850  ∅c0 4282

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 2147  ax-9 2155  ax-ext 2734

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 2741  df-cleq 2754  df-clel 2837  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-nul 4283

This theorem is used by:  disjdsct  33183  onfrALTlem5  45373  onfrALTlem4  45374  onfrALTlem5VD  45715  onfrALTlem4VD  45716