Theorem sbnfc2 3199

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 2816	. . . . 5 ⊢ 𝑦 ∈ V
2		csbtt 3150	. . . . 5 ⊢ ((𝑦 ∈ V ∧ Ⅎ𝑥𝐴) → ⦋𝑦 / 𝑥⦌𝐴 = 𝐴)
3	1, 2	mpan 424	. . . 4 ⊢ (Ⅎ𝑥𝐴 → ⦋𝑦 / 𝑥⦌𝐴 = 𝐴)
4		vex 2816	. . . . 5 ⊢ 𝑧 ∈ V
5		csbtt 3150	. . . . 5 ⊢ ((𝑧 ∈ V ∧ Ⅎ𝑥𝐴) → ⦋𝑧 / 𝑥⦌𝐴 = 𝐴)
6	4, 5	mpan 424	. . . 4 ⊢ (Ⅎ𝑥𝐴 → ⦋𝑧 / 𝑥⦌𝐴 = 𝐴)
7	3, 6	eqtr4d 2268	. . 3 ⊢ (Ⅎ𝑥𝐴 → ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴)
8	7	alrimivv 1924	. 2 ⊢ (Ⅎ𝑥𝐴 → ∀𝑦∀𝑧⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴)
9		nfv 1577	. . 3 ⊢ Ⅎ𝑤∀𝑦∀𝑧⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴
10		eleq2 2296	. . . . . 6 ⊢ (⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴 → (𝑤 ∈ ⦋𝑦 / 𝑥⦌𝐴 ↔ 𝑤 ∈ ⦋𝑧 / 𝑥⦌𝐴))
11		sbsbc 3046	. . . . . . 7 ⊢ ([𝑦 / 𝑥]𝑤 ∈ 𝐴 ↔ [𝑦 / 𝑥]𝑤 ∈ 𝐴)
12		sbcel2g 3159	. . . . . . . 8 ⊢ (𝑦 ∈ V → ([𝑦 / 𝑥]𝑤 ∈ 𝐴 ↔ 𝑤 ∈ ⦋𝑦 / 𝑥⦌𝐴))
13	1, 12	ax-mp 5	. . . . . . 7 ⊢ ([𝑦 / 𝑥]𝑤 ∈ 𝐴 ↔ 𝑤 ∈ ⦋𝑦 / 𝑥⦌𝐴)
14	11, 13	bitri 184	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝑤 ∈ 𝐴 ↔ 𝑤 ∈ ⦋𝑦 / 𝑥⦌𝐴)
15		sbsbc 3046	. . . . . . 7 ⊢ ([𝑧 / 𝑥]𝑤 ∈ 𝐴 ↔ [𝑧 / 𝑥]𝑤 ∈ 𝐴)
16		sbcel2g 3159	. . . . . . . 8 ⊢ (𝑧 ∈ V → ([𝑧 / 𝑥]𝑤 ∈ 𝐴 ↔ 𝑤 ∈ ⦋𝑧 / 𝑥⦌𝐴))
17	4, 16	ax-mp 5	. . . . . . 7 ⊢ ([𝑧 / 𝑥]𝑤 ∈ 𝐴 ↔ 𝑤 ∈ ⦋𝑧 / 𝑥⦌𝐴)
18	15, 17	bitri 184	. . . . . 6 ⊢ ([𝑧 / 𝑥]𝑤 ∈ 𝐴 ↔ 𝑤 ∈ ⦋𝑧 / 𝑥⦌𝐴)
19	10, 14, 18	3bitr4g 223	. . . . 5 ⊢ (⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴 → ([𝑦 / 𝑥]𝑤 ∈ 𝐴 ↔ [𝑧 / 𝑥]𝑤 ∈ 𝐴))
20	19	2alimi 1505	. . . 4 ⊢ (∀𝑦∀𝑧⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴 → ∀𝑦∀𝑧([𝑦 / 𝑥]𝑤 ∈ 𝐴 ↔ [𝑧 / 𝑥]𝑤 ∈ 𝐴))
21		sbnf2 2035	. . . 4 ⊢ (Ⅎ𝑥 𝑤 ∈ 𝐴 ↔ ∀𝑦∀𝑧([𝑦 / 𝑥]𝑤 ∈ 𝐴 ↔ [𝑧 / 𝑥]𝑤 ∈ 𝐴))
22	20, 21	sylibr 134	. . 3 ⊢ (∀𝑦∀𝑧⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴 → Ⅎ𝑥 𝑤 ∈ 𝐴)
23	9, 22	nfcd 2379	. 2 ⊢ (∀𝑦∀𝑧⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴 → Ⅎ𝑥𝐴)
24	8, 23	impbii 126	1 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦∀𝑧⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴)

Syntax hints:   ↔ wb 105  ∀wal 1396   = wceq 1398  Ⅎwnf 1509  [wsb 1811   ∈ wcel 2203  Ⅎwnfc 2371  Vcvv 2813  [wsbc 3042  ⦋csb 3138

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-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-sbc 3043  df-csb 3139

This theorem is referenced by:  eusvnf  4574