Theorem sbnfc2 3196

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 2862	. . . . 5 ⊢ y ∈ V
2		csbtt 3148	. . . . 5 ⊢ ((y ∈ V ∧ ℲxA) → [y / x]A = A)
3	1, 2	mpan 651	. . . 4 ⊢ (ℲxA → [y / x]A = A)
4		vex 2862	. . . . 5 ⊢ z ∈ V
5		csbtt 3148	. . . . 5 ⊢ ((z ∈ V ∧ ℲxA) → [z / x]A = A)
6	4, 5	mpan 651	. . . 4 ⊢ (ℲxA → [z / x]A = A)
7	3, 6	eqtr4d 2388	. . 3 ⊢ (ℲxA → [y / x]A = [z / x]A)
8	7	alrimivv 1632	. 2 ⊢ (ℲxA → ∀y∀z[y / x]A = [z / x]A)
9		nfv 1619	. . 3 ⊢ Ⅎw∀y∀z[y / x]A = [z / x]A
10		eleq2 2414	. . . . . 6 ⊢ ([y / x]A = [z / x]A → (w ∈ [y / x]A ↔ w ∈ [z / x]A))
11		sbsbc 3050	. . . . . . 7 ⊢ ([y / x]w ∈ A ↔ [̣y / x]̣w ∈ A)
12		sbcel2g 3157	. . . . . . . 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 240	. . . . . 6 ⊢ ([y / x]w ∈ A ↔ w ∈ [y / x]A)
15		sbsbc 3050	. . . . . . 7 ⊢ ([z / x]w ∈ A ↔ [̣z / x]̣w ∈ A)
16		sbcel2g 3157	. . . . . . . 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 240	. . . . . 6 ⊢ ([z / x]w ∈ A ↔ w ∈ [z / x]A)
19	10, 14, 18	3bitr4g 279	. . . . 5 ⊢ ([y / x]A = [z / x]A → ([y / x]w ∈ A ↔ [z / x]w ∈ A))
20	19	2alimi 1560	. . . 4 ⊢ (∀y∀z[y / x]A = [z / x]A → ∀y∀z([y / x]w ∈ A ↔ [z / x]w ∈ A))
21		sbnf2 2108	. . . 4 ⊢ (Ⅎx w ∈ A ↔ ∀y∀z([y / x]w ∈ A ↔ [z / x]w ∈ A))
22	20, 21	sylibr 203	. . 3 ⊢ (∀y∀z[y / x]A = [z / x]A → Ⅎx w ∈ A)
23	9, 22	nfcd 2484	. 2 ⊢ (∀y∀z[y / x]A = [z / x]A → ℲxA)
24	8, 23	impbii 180	1 ⊢ (ℲxA ↔ ∀y∀z[y / x]A = [z / x]A)

Syntax hints:   ↔ wb 176  ∀wal 1540  Ⅎwnf 1544   = wceq 1642  [wsb 1648   ∈ wcel 1710  Ⅎwnfc 2476  Vcvv 2859  [̣wsbc 3046  [csb 3136

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-sbc 3047  df-csb 3137

This theorem is referenced by: (None)