Theorem sbcne12 4366

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 2960	. . . . . 6 ⊢ (¬ 𝐵 ≠ 𝐶 ↔ 𝐵 = 𝐶)
2	1	sbcbii 3798	. . . . 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 3789	. . . 4 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥] ¬ 𝐵 ≠ 𝐶 ↔ ¬ [𝐴 / 𝑥]𝐵 ≠ 𝐶))
5		sbceqg 4363	. . . . 5 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶))
6		nne 2960	. . . . 5 ⊢ (¬ ⦋𝐴 / 𝑥⦌𝐵 ≠ ⦋𝐴 / 𝑥⦌𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)
7	5, 6	bitr4di 291	. . . 4 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ¬ ⦋𝐴 / 𝑥⦌𝐵 ≠ ⦋𝐴 / 𝑥⦌𝐶))
8	3, 4, 7	3bitr3d 311	. . 3 ⊢ (𝐴 ∈ V → (¬ [𝐴 / 𝑥]𝐵 ≠ 𝐶 ↔ ¬ ⦋𝐴 / 𝑥⦌𝐵 ≠ ⦋𝐴 / 𝑥⦌𝐶))
9	8	con4bid 319	. 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝐵 ≠ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ≠ ⦋𝐴 / 𝑥⦌𝐶))
10		sbcex 3752	. . . 4 ⊢ ([𝐴 / 𝑥]𝐵 ≠ 𝐶 → 𝐴 ∈ V)
11	10	con3i 154	. . 3 ⊢ (¬ 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝐵 ≠ 𝐶)
12		csbprc 4360	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅)
13		csbprc 4360	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐶 = ∅)
14	12, 13	eqtr4d 2799	. . . 4 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)
15	14, 6	sylibr 236	. . 3 ⊢ (¬ 𝐴 ∈ V → ¬ ⦋𝐴 / 𝑥⦌𝐵 ≠ ⦋𝐴 / 𝑥⦌𝐶)
16	11, 15	2falsed 378	. 2 ⊢ (¬ 𝐴 ∈ V → ([𝐴 / 𝑥]𝐵 ≠ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ≠ ⦋𝐴 / 𝑥⦌𝐶))
17	9, 16	pm2.61i 183	1 ⊢ ([𝐴 / 𝑥]𝐵 ≠ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ≠ ⦋𝐴 / 𝑥⦌𝐶)

Syntax hints:  ¬ wn 3   ↔ wb 208   = wceq 1559   ∈ wcel 2141   ≠ wne 2956  Vcvv 3453  [wsbc 3742  ⦋csb 3850  ∅c0 4283

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-nul 4284

This theorem is referenced by:  2nreu  4395  disjdsct  32865  cdlemkid3N  41517  cdlemkid4  41518