Theorem sbnfc2 3141

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 2791	. . . . 5 |- y e. _V
2		csbtt 3093	. . . . 5 |- ( ( y e. _V /\ F/_ x A ) -> [_ y / x ]_ A = A )
3	1, 2	mpan 651	. . . 4 |- ( F/_ x A -> [_ y / x ]_ A = A )
4		vex 2791	. . . . 5 |- z e. _V
5		csbtt 3093	. . . . 5 |- ( ( z e. _V /\ F/_ x A ) -> [_ z / x ]_ A = A )
6	4, 5	mpan 651	. . . 4 |- ( F/_ x A -> [_ z / x ]_ A = A )
7	3, 6	eqtr4d 2318	. . 3 |- ( F/_ x A -> [_ y / x ]_ A = [_ z / x ]_ A )
8	7	alrimivv 1618	. 2 |- ( F/_ x A -> A. y A. z [_ y / x ]_ A = [_ z / x ]_ A )
9		nfv 1605	. . 3 |- F/ w A. y A. z [_ y / x ]_ A = [_ z / x ]_ A
10		eleq2 2344	. . . . . 6 |- ( [_ y / x ]_ A = [_ z / x ]_ A -> ( w e. [_ y / x ]_ A <-> w e. [_ z / x ]_ A ) )
11		sbsbc 2995	. . . . . . 7 |- ( [ y / x ] w e. A <-> [. y / x ]. w e. A )
12		sbcel2g 3102	. . . . . . . 8 |- ( y e. _V -> ( [. y / x ]. w e. A <-> w e. [_ y / x ]_ A ) )
13	1, 12	ax-mp 8	. . . . . . 7 |- ( [. y / x ]. w e. A <-> w e. [_ y / x ]_ A )
14	11, 13	bitri 240	. . . . . 6 |- ( [ y / x ] w e. A <-> w e. [_ y / x ]_ A )
15		sbsbc 2995	. . . . . . 7 |- ( [ z / x ] w e. A <-> [. z / x ]. w e. A )
16		sbcel2g 3102	. . . . . . . 8 |- ( z e. _V -> ( [. z / x ]. w e. A <-> w e. [_ z / x ]_ A ) )
17	4, 16	ax-mp 8	. . . . . . 7 |- ( [. z / x ]. w e. A <-> w e. [_ z / x ]_ A )
18	15, 17	bitri 240	. . . . . 6 |- ( [ z / x ] w e. A <-> w e. [_ z / x ]_ A )
19	10, 14, 18	3bitr4g 279	. . . . 5 |- ( [_ y / x ]_ A = [_ z / x ]_ A -> ( [ y / x ] w e. A <-> [ z / x ] w e. A ) )
20	19	2alimi 1547	. . . 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 2047	. . . 4 |- ( F/ x w e. A <-> A. y A. z ( [ y / x ] w e. A <-> [ z / x ] w e. A ) )
22	20, 21	sylibr 203	. . 3 |- ( A. y A. z [_ y / x ]_ A = [_ z / x ]_ A -> F/ x w e. A )
23	9, 22	nfcd 2414	. 2 |- ( A. y A. z [_ y / x ]_ A = [_ z / x ]_ A -> F/_ x A )
24	8, 23	impbii 180	1 |- ( F/_ x A <-> A. y A. z [_ y / x ]_ A = [_ z / x ]_ A )

Syntax hints:   <-> wb 176   A.wal 1527   F/wnf 1531   = wceq 1623   [wsb 1629   e. wcel 1684   F/_wnfc 2406   _Vcvv 2788   [.wsbc 2991   [_csb 3081

This theorem is referenced by:  eusvnf  4529

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-sbc 2992  df-csb 3082