Theorem sbcne12g 3063

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

Assertion

Ref	Expression
sbcne12g	⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 ≠ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ≠ ⦋𝐴 / 𝑥⦌𝐶))

Proof of Theorem sbcne12g

Step	Hyp	Ref	Expression
1		sbceqg 3061	. . 3 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶))
2	1	notbid 657	. 2 ⊢ (𝐴 ∈ 𝑉 → (¬ [𝐴 / 𝑥]𝐵 = 𝐶 ↔ ¬ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶))
3		df-ne 2337	. . . . 5 ⊢ (𝐵 ≠ 𝐶 ↔ ¬ 𝐵 = 𝐶)
4	3	sbcbii 3010	. . . 4 ⊢ ([𝐴 / 𝑥]𝐵 ≠ 𝐶 ↔ [𝐴 / 𝑥] ¬ 𝐵 = 𝐶)
5		sbcng 2991	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥] ¬ 𝐵 = 𝐶 ↔ ¬ [𝐴 / 𝑥]𝐵 = 𝐶))
6	4, 5	syl5bb 191	. . 3 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 ≠ 𝐶 ↔ ¬ [𝐴 / 𝑥]𝐵 = 𝐶))
7		df-ne 2337	. . . 4 ⊢ (⦋𝐴 / 𝑥⦌𝐵 ≠ ⦋𝐴 / 𝑥⦌𝐶 ↔ ¬ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)
8	7	a1i 9	. . 3 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝐵 ≠ ⦋𝐴 / 𝑥⦌𝐶 ↔ ¬ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶))
9	6, 8	bibi12d 234	. 2 ⊢ (𝐴 ∈ 𝑉 → (([𝐴 / 𝑥]𝐵 ≠ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ≠ ⦋𝐴 / 𝑥⦌𝐶) ↔ (¬ [𝐴 / 𝑥]𝐵 = 𝐶 ↔ ¬ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)))
10	2, 9	mpbird 166	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 ≠ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ≠ ⦋𝐴 / 𝑥⦌𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104   = wceq 1343   ∈ wcel 2136   ≠ wne 2336  [wsbc 2951  ⦋csb 3045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-v 2728  df-sbc 2952  df-csb 3046

This theorem is referenced by: (None)