Theorem sbcnel12g 4342

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		sbcng 3770	. 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥] ¬ 𝐵 ∈ 𝐶 ↔ ¬ [𝐴 / 𝑥]𝐵 ∈ 𝐶))
2		df-nel 3039	. . 3 ⊢ (𝐵 ∉ 𝐶 ↔ ¬ 𝐵 ∈ 𝐶)
3	2	sbcbii 3779	. 2 ⊢ ([𝐴 / 𝑥]𝐵 ∉ 𝐶 ↔ [𝐴 / 𝑥] ¬ 𝐵 ∈ 𝐶)
4		df-nel 3039	. . 3 ⊢ (⦋𝐴 / 𝑥⦌𝐵 ∉ ⦋𝐴 / 𝑥⦌𝐶 ↔ ¬ ⦋𝐴 / 𝑥⦌𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶)
5		sbcel12 4339	. . 3 ⊢ ([𝐴 / 𝑥]𝐵 ∈ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶)
6	4, 5	xchbinxr 336	. 2 ⊢ (⦋𝐴 / 𝑥⦌𝐵 ∉ ⦋𝐴 / 𝑥⦌𝐶 ↔ ¬ [𝐴 / 𝑥]𝐵 ∈ 𝐶)
7	1, 3, 6	3bitr4g 315	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 ∉ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ∉ ⦋𝐴 / 𝑥⦌𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∈ wcel 2119   ∉ wnel 3038  [wsbc 3723  ⦋csb 3831

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-nel 3039  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-nul 4262

This theorem is referenced by:  rusbcALT  44882