Theorem sbnfc2 3959

Description: Two ways of expressing "𝑥 is (effectively) not free in 𝐴." (Contributed by Mario Carneiro, 14-Oct-2016.)

Assertion

Ref	Expression
sbnfc2	⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦∀𝑧⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴)

Distinct variable groups:   𝑥,𝑦,𝑧   𝑦,𝐴,𝑧

Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem sbnfc2

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3176	. . . . 5 ⊢ 𝑦 ∈ V
2		csbtt 3510	. . . . 5 ⊢ ((𝑦 ∈ V ∧ Ⅎ𝑥𝐴) → ⦋𝑦 / 𝑥⦌𝐴 = 𝐴)
3	1, 2	mpan 702	. . . 4 ⊢ (Ⅎ𝑥𝐴 → ⦋𝑦 / 𝑥⦌𝐴 = 𝐴)
4		vex 3176	. . . . 5 ⊢ 𝑧 ∈ V
5		csbtt 3510	. . . . 5 ⊢ ((𝑧 ∈ V ∧ Ⅎ𝑥𝐴) → ⦋𝑧 / 𝑥⦌𝐴 = 𝐴)
6	4, 5	mpan 702	. . . 4 ⊢ (Ⅎ𝑥𝐴 → ⦋𝑧 / 𝑥⦌𝐴 = 𝐴)
7	3, 6	eqtr4d 2647	. . 3 ⊢ (Ⅎ𝑥𝐴 → ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴)
8	7	alrimivv 1843	. 2 ⊢ (Ⅎ𝑥𝐴 → ∀𝑦∀𝑧⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴)
9		nfv 1830	. . 3 ⊢ Ⅎ𝑤∀𝑦∀𝑧⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴
10		eleq2 2677	. . . . . 6 ⊢ (⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴 → (𝑤 ∈ ⦋𝑦 / 𝑥⦌𝐴 ↔ 𝑤 ∈ ⦋𝑧 / 𝑥⦌𝐴))
11		sbsbc 3406	. . . . . . 7 ⊢ ([𝑦 / 𝑥]𝑤 ∈ 𝐴 ↔ [𝑦 / 𝑥]𝑤 ∈ 𝐴)
12		sbcel2 3941	. . . . . . 7 ⊢ ([𝑦 / 𝑥]𝑤 ∈ 𝐴 ↔ 𝑤 ∈ ⦋𝑦 / 𝑥⦌𝐴)
13	11, 12	bitri 263	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝑤 ∈ 𝐴 ↔ 𝑤 ∈ ⦋𝑦 / 𝑥⦌𝐴)
14		sbsbc 3406	. . . . . . 7 ⊢ ([𝑧 / 𝑥]𝑤 ∈ 𝐴 ↔ [𝑧 / 𝑥]𝑤 ∈ 𝐴)
15		sbcel2 3941	. . . . . . 7 ⊢ ([𝑧 / 𝑥]𝑤 ∈ 𝐴 ↔ 𝑤 ∈ ⦋𝑧 / 𝑥⦌𝐴)
16	14, 15	bitri 263	. . . . . 6 ⊢ ([𝑧 / 𝑥]𝑤 ∈ 𝐴 ↔ 𝑤 ∈ ⦋𝑧 / 𝑥⦌𝐴)
17	10, 13, 16	3bitr4g 302	. . . . 5 ⊢ (⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴 → ([𝑦 / 𝑥]𝑤 ∈ 𝐴 ↔ [𝑧 / 𝑥]𝑤 ∈ 𝐴))
18	17	2alimi 1731	. . . 4 ⊢ (∀𝑦∀𝑧⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴 → ∀𝑦∀𝑧([𝑦 / 𝑥]𝑤 ∈ 𝐴 ↔ [𝑧 / 𝑥]𝑤 ∈ 𝐴))
19		sbnf2 2427	. . . 4 ⊢ (Ⅎ𝑥 𝑤 ∈ 𝐴 ↔ ∀𝑦∀𝑧([𝑦 / 𝑥]𝑤 ∈ 𝐴 ↔ [𝑧 / 𝑥]𝑤 ∈ 𝐴))
20	18, 19	sylibr 223	. . 3 ⊢ (∀𝑦∀𝑧⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴 → Ⅎ𝑥 𝑤 ∈ 𝐴)
21	9, 20	nfcd 2746	. 2 ⊢ (∀𝑦∀𝑧⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴 → Ⅎ𝑥𝐴)
22	8, 21	impbii 198	1 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦∀𝑧⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴)

Syntax hints:   ↔ wb 195  ∀wal 1473   = wceq 1475  Ⅎwnf 1699  [wsb 1867   ∈ wcel 1977  Ⅎwnfc 2738  Vcvv 3173  [wsbc 3402  ⦋csb 3499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-nul 3875

This theorem is referenced by:  eusvnf  4787