Theorem sbcnel12g 3074

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 2443	. . . 4 ⊢ (𝐵 ∉ 𝐶 ↔ ¬ 𝐵 ∈ 𝐶)
2	1	sbcbii 3022	. . 3 ⊢ ([𝐴 / 𝑥]𝐵 ∉ 𝐶 ↔ [𝐴 / 𝑥] ¬ 𝐵 ∈ 𝐶)
3	2	a1i 9	. 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 ∉ 𝐶 ↔ [𝐴 / 𝑥] ¬ 𝐵 ∈ 𝐶))
4		sbcng 3003	. 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥] ¬ 𝐵 ∈ 𝐶 ↔ ¬ [𝐴 / 𝑥]𝐵 ∈ 𝐶))
5		sbcel12g 3072	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 ∈ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶))
6	5	notbid 667	. . 3 ⊢ (𝐴 ∈ 𝑉 → (¬ [𝐴 / 𝑥]𝐵 ∈ 𝐶 ↔ ¬ ⦋𝐴 / 𝑥⦌𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶))
7		df-nel 2443	. . 3 ⊢ (⦋𝐴 / 𝑥⦌𝐵 ∉ ⦋𝐴 / 𝑥⦌𝐶 ↔ ¬ ⦋𝐴 / 𝑥⦌𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶)
8	6, 7	bitr4di 198	. 2 ⊢ (𝐴 ∈ 𝑉 → (¬ [𝐴 / 𝑥]𝐵 ∈ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ∉ ⦋𝐴 / 𝑥⦌𝐶))
9	3, 4, 8	3bitrd 214	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 ∉ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ∉ ⦋𝐴 / 𝑥⦌𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105   ∈ wcel 2148   ∉ wnel 2442  [wsbc 2962  ⦋csb 3057

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-nel 2443  df-v 2739  df-sbc 2963  df-csb 3058

This theorem is referenced by: (None)