Theorem sbnfc2 3208

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 2824	. . . . 5 ⊢ 𝑦 ∈ V
2		csbtt 3159	. . . . 5 ⊢ ((𝑦 ∈ V ∧ Ⅎ𝑥𝐴) → ⦋𝑦 / 𝑥⦌𝐴 = 𝐴)
3	1, 2	mpan 428	. . . 4 ⊢ (Ⅎ𝑥𝐴 → ⦋𝑦 / 𝑥⦌𝐴 = 𝐴)
4		vex 2824	. . . . 5 ⊢ 𝑧 ∈ V
5		csbtt 3159	. . . . 5 ⊢ ((𝑧 ∈ V ∧ Ⅎ𝑥𝐴) → ⦋𝑧 / 𝑥⦌𝐴 = 𝐴)
6	4, 5	mpan 428	. . . 4 ⊢ (Ⅎ𝑥𝐴 → ⦋𝑧 / 𝑥⦌𝐴 = 𝐴)
7	3, 6	eqtr4d 2274	. . 3 ⊢ (Ⅎ𝑥𝐴 → ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴)
8	7	alrimivv 1928	. 2 ⊢ (Ⅎ𝑥𝐴 → ∀𝑦∀𝑧⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴)
9		nfv 1581	. . 3 ⊢ Ⅎ𝑤∀𝑦∀𝑧⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴
10		eleq2 2302	. . . . . 6 ⊢ (⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴 → (𝑤 ∈ ⦋𝑦 / 𝑥⦌𝐴 ↔ 𝑤 ∈ ⦋𝑧 / 𝑥⦌𝐴))
11		sbsbc 3055	. . . . . . 7 ⊢ ([𝑦 / 𝑥]𝑤 ∈ 𝐴 ↔ [𝑦 / 𝑥]𝑤 ∈ 𝐴)
12		sbcel2g 3168	. . . . . . . 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 3055	. . . . . . 7 ⊢ ([𝑧 / 𝑥]𝑤 ∈ 𝐴 ↔ [𝑧 / 𝑥]𝑤 ∈ 𝐴)
16		sbcel2g 3168	. . . . . . . 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 1509	. . . 4 ⊢ (∀𝑦∀𝑧⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴 → ∀𝑦∀𝑧([𝑦 / 𝑥]𝑤 ∈ 𝐴 ↔ [𝑧 / 𝑥]𝑤 ∈ 𝐴))
21		sbnf2 2041	. . . 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 2387	. 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 1400   = wceq 1402  Ⅎwnf 1513  [wsb 1815   ∈ wcel 2209  Ⅎwnfc 2379  Vcvv 2821  [wsbc 3051  ⦋csb 3147

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-sbc 3052  df-csb 3148

This theorem is referenced by:  eusvnf  4594