Theorem sbcne12 4386

Description: Distribute proper substitution through an inequality. (Contributed by Andrew Salmon, 18-Jun-2011.) (Revised by NM, 18-Aug-2018.)

Assertion

Ref	Expression
sbcne12	⊢ ([𝐴 / 𝑥]𝐵 ≠ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ≠ ⦋𝐴 / 𝑥⦌𝐶)

Proof of Theorem sbcne12

Step	Hyp	Ref	Expression
1		nne 2931	. . . . . 6 ⊢ (¬ 𝐵 ≠ 𝐶 ↔ 𝐵 = 𝐶)
2	1	sbcbii 3818	. . . . 5 ⊢ ([𝐴 / 𝑥] ¬ 𝐵 ≠ 𝐶 ↔ [𝐴 / 𝑥]𝐵 = 𝐶)
3	2	a1i 11	. . . 4 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥] ¬ 𝐵 ≠ 𝐶 ↔ [𝐴 / 𝑥]𝐵 = 𝐶))
4		sbcng 3809	. . . 4 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥] ¬ 𝐵 ≠ 𝐶 ↔ ¬ [𝐴 / 𝑥]𝐵 ≠ 𝐶))
5		sbceqg 4383	. . . . 5 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶))
6		nne 2931	. . . . 5 ⊢ (¬ ⦋𝐴 / 𝑥⦌𝐵 ≠ ⦋𝐴 / 𝑥⦌𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)
7	5, 6	bitr4di 289	. . . 4 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ¬ ⦋𝐴 / 𝑥⦌𝐵 ≠ ⦋𝐴 / 𝑥⦌𝐶))
8	3, 4, 7	3bitr3d 309	. . 3 ⊢ (𝐴 ∈ V → (¬ [𝐴 / 𝑥]𝐵 ≠ 𝐶 ↔ ¬ ⦋𝐴 / 𝑥⦌𝐵 ≠ ⦋𝐴 / 𝑥⦌𝐶))
9	8	con4bid 317	. 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝐵 ≠ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ≠ ⦋𝐴 / 𝑥⦌𝐶))
10		sbcex 3771	. . . 4 ⊢ ([𝐴 / 𝑥]𝐵 ≠ 𝐶 → 𝐴 ∈ V)
11	10	con3i 154	. . 3 ⊢ (¬ 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝐵 ≠ 𝐶)
12		csbprc 4380	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅)
13		csbprc 4380	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐶 = ∅)
14	12, 13	eqtr4d 2768	. . . 4 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)
15	14, 6	sylibr 234	. . 3 ⊢ (¬ 𝐴 ∈ V → ¬ ⦋𝐴 / 𝑥⦌𝐵 ≠ ⦋𝐴 / 𝑥⦌𝐶)
16	11, 15	2falsed 376	. 2 ⊢ (¬ 𝐴 ∈ V → ([𝐴 / 𝑥]𝐵 ≠ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ≠ ⦋𝐴 / 𝑥⦌𝐶))
17	9, 16	pm2.61i 182	1 ⊢ ([𝐴 / 𝑥]𝐵 ≠ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ≠ ⦋𝐴 / 𝑥⦌𝐶)

Syntax hints:  ¬ wn 3   ↔ wb 206   = wceq 1540   ∈ wcel 2109   ≠ wne 2927  Vcvv 3455  [wsbc 3761  ⦋csb 3870  ∅c0 4304

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2880  df-ne 2928  df-v 3457  df-sbc 3762  df-csb 3871  df-dif 3925  df-nul 4305

This theorem is referenced by:  2nreu  4415  disjdsct  32634  cdlemkid3N  40919  cdlemkid4  40920