Theorem sbcne12g 3154

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		nne 2520	. . . . 5 ⊢ (¬ B ≠ C ↔ B = C)
2	1	sbcbii 3101	. . . 4 ⊢ ([̣A / x]̣ ¬ B ≠ C ↔ [̣A / x]̣B = C)
3	2	a1i 10	. . 3 ⊢ (A ∈ V → ([̣A / x]̣ ¬ B ≠ C ↔ [̣A / x]̣B = C))
4		sbcng 3086	. . 3 ⊢ (A ∈ V → ([̣A / x]̣ ¬ B ≠ C ↔ ¬ [̣A / x]̣B ≠ C))
5		sbceqg 3152	. . . 4 ⊢ (A ∈ V → ([̣A / x]̣B = C ↔ [A / x]B = [A / x]C))
6		nne 2520	. . . 4 ⊢ (¬ [A / x]B ≠ [A / x]C ↔ [A / x]B = [A / x]C)
7	5, 6	syl6bbr 254	. . 3 ⊢ (A ∈ V → ([̣A / x]̣B = C ↔ ¬ [A / x]B ≠ [A / x]C))
8	3, 4, 7	3bitr3d 274	. 2 ⊢ (A ∈ V → (¬ [̣A / x]̣B ≠ C ↔ ¬ [A / x]B ≠ [A / x]C))
9	8	con4bid 284	1 ⊢ (A ∈ V → ([̣A / x]̣B ≠ C ↔ [A / x]B ≠ [A / x]C))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  [̣wsbc 3046  [csb 3136

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-v 2861  df-sbc 3047  df-csb 3137

This theorem is referenced by: (None)