Theorem sbnfc2 3065

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 2692	. . . . 5 |- y e. _V
2		csbtt 3019	. . . . 5 |- ( ( y e. _V /\ F/_ x A ) -> [_ y / x ]_ A = A )
3	1, 2	mpan 421	. . . 4 |- ( F/_ x A -> [_ y / x ]_ A = A )
4		vex 2692	. . . . 5 |- z e. _V
5		csbtt 3019	. . . . 5 |- ( ( z e. _V /\ F/_ x A ) -> [_ z / x ]_ A = A )
6	4, 5	mpan 421	. . . 4 |- ( F/_ x A -> [_ z / x ]_ A = A )
7	3, 6	eqtr4d 2176	. . 3 |- ( F/_ x A -> [_ y / x ]_ A = [_ z / x ]_ A )
8	7	alrimivv 1848	. 2 |- ( F/_ x A -> A. y A. z [_ y / x ]_ A = [_ z / x ]_ A )
9		nfv 1509	. . 3 |- F/ w A. y A. z [_ y / x ]_ A = [_ z / x ]_ A
10		eleq2 2204	. . . . . 6 |- ( [_ y / x ]_ A = [_ z / x ]_ A -> ( w e. [_ y / x ]_ A <-> w e. [_ z / x ]_ A ) )
11		sbsbc 2917	. . . . . . 7 |- ( [ y / x ] w e. A <-> [. y / x ]. w e. A )
12		sbcel2g 3028	. . . . . . . 8 |- ( y e. _V -> ( [. y / x ]. w e. A <-> w e. [_ y / x ]_ A ) )
13	1, 12	ax-mp 5	. . . . . . 7 |- ( [. y / x ]. w e. A <-> w e. [_ y / x ]_ A )
14	11, 13	bitri 183	. . . . . 6 |- ( [ y / x ] w e. A <-> w e. [_ y / x ]_ A )
15		sbsbc 2917	. . . . . . 7 |- ( [ z / x ] w e. A <-> [. z / x ]. w e. A )
16		sbcel2g 3028	. . . . . . . 8 |- ( z e. _V -> ( [. z / x ]. w e. A <-> w e. [_ z / x ]_ A ) )
17	4, 16	ax-mp 5	. . . . . . 7 |- ( [. z / x ]. w e. A <-> w e. [_ z / x ]_ A )
18	15, 17	bitri 183	. . . . . 6 |- ( [ z / x ] w e. A <-> w e. [_ z / x ]_ A )
19	10, 14, 18	3bitr4g 222	. . . . 5 |- ( [_ y / x ]_ A = [_ z / x ]_ A -> ( [ y / x ] w e. A <-> [ z / x ] w e. A ) )
20	19	2alimi 1433	. . . 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 1957	. . . 4 |- ( F/ x w e. A <-> A. y A. z ( [ y / x ] w e. A <-> [ z / x ] w e. A ) )
22	20, 21	sylibr 133	. . 3 |- ( A. y A. z [_ y / x ]_ A = [_ z / x ]_ A -> F/ x w e. A )
23	9, 22	nfcd 2277	. 2 |- ( A. y A. z [_ y / x ]_ A = [_ z / x ]_ A -> F/_ x A )
24	8, 23	impbii 125	1 |- ( F/_ x A <-> A. y A. z [_ y / x ]_ A = [_ z / x ]_ A )

Syntax hints:   <-> wb 104   A.wal 1330   = wceq 1332   F/wnf 1437   e. wcel 1481   [wsb 1736   F/_wnfc 2269   _Vcvv 2689   [.wsbc 2913   [_csb 3007

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-sbc 2914  df-csb 3008

This theorem is referenced by:  eusvnf  4382