Theorem sbcnel12g 3154

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

Assertion

Ref	Expression
sbcnel12g	⊢ (A ∈ V → ([̣A / x]̣B ∉ C ↔ [A / x]B ∉ [A / x]C))

Proof of Theorem sbcnel12g

Step	Hyp	Ref	Expression
1		df-nel 2520	. . . 4 ⊢ (B ∉ C ↔ ¬ B ∈ C)
2	1	sbcbii 3102	. . 3 ⊢ ([̣A / x]̣B ∉ C ↔ [̣A / x]̣ ¬ B ∈ C)
3	2	a1i 10	. 2 ⊢ (A ∈ V → ([̣A / x]̣B ∉ C ↔ [̣A / x]̣ ¬ B ∈ C))
4		sbcng 3087	. 2 ⊢ (A ∈ V → ([̣A / x]̣ ¬ B ∈ C ↔ ¬ [̣A / x]̣B ∈ C))
5		sbcel12g 3152	. . . 4 ⊢ (A ∈ V → ([̣A / x]̣B ∈ C ↔ [A / x]B ∈ [A / x]C))
6	5	notbid 285	. . 3 ⊢ (A ∈ V → (¬ [̣A / x]̣B ∈ C ↔ ¬ [A / x]B ∈ [A / x]C))
7		df-nel 2520	. . 3 ⊢ ([A / x]B ∉ [A / x]C ↔ ¬ [A / x]B ∈ [A / x]C)
8	6, 7	syl6bbr 254	. 2 ⊢ (A ∈ V → (¬ [̣A / x]̣B ∈ C ↔ [A / x]B ∉ [A / x]C))
9	3, 4, 8	3bitrd 270	1 ⊢ (A ∈ V → ([̣A / x]̣B ∉ C ↔ [A / x]B ∉ [A / x]C))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∈ wcel 1710   ∉ wnel 2518  [̣wsbc 3047  [csb 3137

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 2479  df-nel 2520  df-v 2862  df-sbc 3048  df-csb 3138

This theorem is referenced by: (None)