Theorem sbnfc2 4414

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 3463	. . . . 5 ⊢ 𝑦 ∈ V
2		csbtt 3891	. . . . 5 ⊢ ((𝑦 ∈ V ∧ Ⅎ𝑥𝐴) → ⦋𝑦 / 𝑥⦌𝐴 = 𝐴)
3	1, 2	mpan 690	. . . 4 ⊢ (Ⅎ𝑥𝐴 → ⦋𝑦 / 𝑥⦌𝐴 = 𝐴)
4		vex 3463	. . . . 5 ⊢ 𝑧 ∈ V
5		csbtt 3891	. . . . 5 ⊢ ((𝑧 ∈ V ∧ Ⅎ𝑥𝐴) → ⦋𝑧 / 𝑥⦌𝐴 = 𝐴)
6	4, 5	mpan 690	. . . 4 ⊢ (Ⅎ𝑥𝐴 → ⦋𝑧 / 𝑥⦌𝐴 = 𝐴)
7	3, 6	eqtr4d 2773	. . 3 ⊢ (Ⅎ𝑥𝐴 → ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴)
8	7	alrimivv 1928	. 2 ⊢ (Ⅎ𝑥𝐴 → ∀𝑦∀𝑧⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴)
9		nfv 1914	. . 3 ⊢ Ⅎ𝑤∀𝑦∀𝑧⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴
10		eleq2 2823	. . . . . 6 ⊢ (⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴 → (𝑤 ∈ ⦋𝑦 / 𝑥⦌𝐴 ↔ 𝑤 ∈ ⦋𝑧 / 𝑥⦌𝐴))
11		sbsbc 3769	. . . . . . 7 ⊢ ([𝑦 / 𝑥]𝑤 ∈ 𝐴 ↔ [𝑦 / 𝑥]𝑤 ∈ 𝐴)
12		sbcel2 4393	. . . . . . 7 ⊢ ([𝑦 / 𝑥]𝑤 ∈ 𝐴 ↔ 𝑤 ∈ ⦋𝑦 / 𝑥⦌𝐴)
13	11, 12	bitri 275	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝑤 ∈ 𝐴 ↔ 𝑤 ∈ ⦋𝑦 / 𝑥⦌𝐴)
14		sbsbc 3769	. . . . . . 7 ⊢ ([𝑧 / 𝑥]𝑤 ∈ 𝐴 ↔ [𝑧 / 𝑥]𝑤 ∈ 𝐴)
15		sbcel2 4393	. . . . . . 7 ⊢ ([𝑧 / 𝑥]𝑤 ∈ 𝐴 ↔ 𝑤 ∈ ⦋𝑧 / 𝑥⦌𝐴)
16	14, 15	bitri 275	. . . . . 6 ⊢ ([𝑧 / 𝑥]𝑤 ∈ 𝐴 ↔ 𝑤 ∈ ⦋𝑧 / 𝑥⦌𝐴)
17	10, 13, 16	3bitr4g 314	. . . . 5 ⊢ (⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴 → ([𝑦 / 𝑥]𝑤 ∈ 𝐴 ↔ [𝑧 / 𝑥]𝑤 ∈ 𝐴))
18	17	2alimi 1812	. . . 4 ⊢ (∀𝑦∀𝑧⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴 → ∀𝑦∀𝑧([𝑦 / 𝑥]𝑤 ∈ 𝐴 ↔ [𝑧 / 𝑥]𝑤 ∈ 𝐴))
19		sbnf2 2360	. . . 4 ⊢ (Ⅎ𝑥 𝑤 ∈ 𝐴 ↔ ∀𝑦∀𝑧([𝑦 / 𝑥]𝑤 ∈ 𝐴 ↔ [𝑧 / 𝑥]𝑤 ∈ 𝐴))
20	18, 19	sylibr 234	. . 3 ⊢ (∀𝑦∀𝑧⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴 → Ⅎ𝑥 𝑤 ∈ 𝐴)
21	9, 20	nfcd 2891	. 2 ⊢ (∀𝑦∀𝑧⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴 → Ⅎ𝑥𝐴)
22	8, 21	impbii 209	1 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦∀𝑧⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴)

Syntax hints:   ↔ wb 206  ∀wal 1538   = wceq 1540  Ⅎwnf 1783  [wsb 2064   ∈ wcel 2108  Ⅎwnfc 2883  Vcvv 3459  [wsbc 3765  ⦋csb 3874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-nul 4309

This theorem is referenced by:  eusvnf  5362