Theorem csbprc 3496

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.)

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		df-csb 3085	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵}
2		sbcex 2998	. . . . . . 7 ⊢ ([𝐴 / 𝑥]𝑦 ∈ 𝐵 → 𝐴 ∈ V)
3	2	con3i 633	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝑦 ∈ 𝐵)
4	3	pm2.21d 620	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ([𝐴 / 𝑥]𝑦 ∈ 𝐵 → ⊥))
5		falim 1378	. . . . 5 ⊢ (⊥ → [𝐴 / 𝑥]𝑦 ∈ 𝐵)
6	4, 5	impbid1 142	. . . 4 ⊢ (¬ 𝐴 ∈ V → ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ ⊥))
7	6	abbidv 2314	. . 3 ⊢ (¬ 𝐴 ∈ V → {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ ⊥})
8		fal 1371	. . . 4 ⊢ ¬ ⊥
9	8	abf 3494	. . 3 ⊢ {𝑦 ∣ ⊥} = ∅
10	7, 9	eqtrdi 2245	. 2 ⊢ (¬ 𝐴 ∈ V → {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} = ∅)
11	1, 10	eqtrid 2241	1 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1364  ⊥wfal 1369   ∈ wcel 2167  {cab 2182  Vcvv 2763  [wsbc 2989  ⦋csb 3084  ∅c0 3450

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-in 3163  df-ss 3170  df-nul 3451

This theorem is referenced by: (None)