Theorem sbnfc2 2962

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

Assertion

Ref	Expression
sbnfc2	|- ( F/_ x A <-> A. y A. 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 2604	. . . . 5 |- y e. _V
2		csbtt 2918	. . . . 5 |- ( ( y e. _V /\ F/_ x A ) -> [_ y / x ]_ A = A )
3	1, 2	mpan 414	. . . 4 |- ( F/_ x A -> [_ y / x ]_ A = A )
4		vex 2604	. . . . 5 |- z e. _V
5		csbtt 2918	. . . . 5 |- ( ( z e. _V /\ F/_ x A ) -> [_ z / x ]_ A = A )
6	4, 5	mpan 414	. . . 4 |- ( F/_ x A -> [_ z / x ]_ A = A )
7	3, 6	eqtr4d 2116	. . 3 |- ( F/_ x A -> [_ y / x ]_ A = [_ z / x ]_ A )
8	7	alrimivv 1796	. 2 |- ( F/_ x A -> A. y A. z [_ y / x ]_ A = [_ z / x ]_ A )
9		nfv 1461	. . 3 |- F/ w A. y A. z [_ y / x ]_ A = [_ z / x ]_ A
10		eleq2 2142	. . . . . 6 |- ( [_ y / x ]_ A = [_ z / x ]_ A -> ( w e. [_ y / x ]_ A <-> w e. [_ z / x ]_ A ) )
11		sbsbc 2819	. . . . . . 7 |- ( [ y / x ] w e. A <-> [. y / x ]. w e. A )
12		sbcel2g 2927	. . . . . . . 8 |- ( y e. _V -> ( [. y / x ]. w e. A <-> w e. [_ y / x ]_ A ) )
13	1, 12	ax-mp 7	. . . . . . 7 |- ( [. y / x ]. w e. A <-> w e. [_ y / x ]_ A )
14	11, 13	bitri 182	. . . . . 6 |- ( [ y / x ] w e. A <-> w e. [_ y / x ]_ A )
15		sbsbc 2819	. . . . . . 7 |- ( [ z / x ] w e. A <-> [. z / x ]. w e. A )
16		sbcel2g 2927	. . . . . . . 8 |- ( z e. _V -> ( [. z / x ]. w e. A <-> w e. [_ z / x ]_ A ) )
17	4, 16	ax-mp 7	. . . . . . 7 |- ( [. z / x ]. w e. A <-> w e. [_ z / x ]_ A )
18	15, 17	bitri 182	. . . . . 6 |- ( [ z / x ] w e. A <-> w e. [_ z / x ]_ A )
19	10, 14, 18	3bitr4g 221	. . . . 5 |- ( [_ y / x ]_ A = [_ z / x ]_ A -> ( [ y / x ] w e. A <-> [ z / x ] w e. A ) )
20	19	2alimi 1385	. . . 4 |- ( A. y A. z [_ y / x ]_ A = [_ z / x ]_ A -> A. y A. z ( [ y / x ] w e. A <-> [ z / x ] w e. A ) )
21		sbnf2 1898	. . . 4 |- ( F/ x w e. A <-> A. y A. z ( [ y / x ] w e. A <-> [ z / x ] w e. A ) )
22	20, 21	sylibr 132	. . 3 |- ( A. y A. z [_ y / x ]_ A = [_ z / x ]_ A -> F/ x w e. A )
23	9, 22	nfcd 2214	. 2 |- ( A. y A. z [_ y / x ]_ A = [_ z / x ]_ A -> F/_ x A )
24	8, 23	impbii 124	1 |- ( F/_ x A <-> A. y A. z [_ y / x ]_ A = [_ z / x ]_ A )

Syntax hints:   <-> wb 103   A.wal 1282   = wceq 1284   F/wnf 1389   e. wcel 1433   [wsb 1685   F/_wnfc 2206   _Vcvv 2601   [.wsbc 2815   [_csb 2908

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-sbc 2816  df-csb 2909

This theorem is referenced by:  eusvnf  4203