Theorem csb0 4359

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 3865	. 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌∅ = ∅)
2		csbprc 4358	. 2 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌∅ = ∅)
3	1, 2	pm2.61i 182	1 ⊢ ⦋𝐴 / 𝑥⦌∅ = ∅

Syntax hints:   = wceq 1541   ∈ wcel 2113  Vcvv 3437  ⦋csb 3846  ∅c0 4282

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-nul 4283

This theorem is referenced by:  disjdsct  32688  onfrALTlem5  44659  onfrALTlem4  44660  onfrALTlem5VD  45001  onfrALTlem4VD  45002