Theorem sbnfc2 3154

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 2775	. . . . 5 |- y e. _V
2		csbtt 3105	. . . . 5 |- ( ( y e. _V /\ F/_ x A ) -> [_ y / x ]_ A = A )
3	1, 2	mpan 424	. . . 4 |- ( F/_ x A -> [_ y / x ]_ A = A )
4		vex 2775	. . . . 5 |- z e. _V
5		csbtt 3105	. . . . 5 |- ( ( z e. _V /\ F/_ x A ) -> [_ z / x ]_ A = A )
6	4, 5	mpan 424	. . . 4 |- ( F/_ x A -> [_ z / x ]_ A = A )
7	3, 6	eqtr4d 2241	. . 3 |- ( F/_ x A -> [_ y / x ]_ A = [_ z / x ]_ A )
8	7	alrimivv 1898	. 2 |- ( F/_ x A -> A. y A. z [_ y / x ]_ A = [_ z / x ]_ A )
9		nfv 1551	. . 3 |- F/ w A. y A. z [_ y / x ]_ A = [_ z / x ]_ A
10		eleq2 2269	. . . . . 6 |- ( [_ y / x ]_ A = [_ z / x ]_ A -> ( w e. [_ y / x ]_ A <-> w e. [_ z / x ]_ A ) )
11		sbsbc 3002	. . . . . . 7 |- ( [ y / x ] w e. A <-> [. y / x ]. w e. A )
12		sbcel2g 3114	. . . . . . . 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 184	. . . . . 6 |- ( [ y / x ] w e. A <-> w e. [_ y / x ]_ A )
15		sbsbc 3002	. . . . . . 7 |- ( [ z / x ] w e. A <-> [. z / x ]. w e. A )
16		sbcel2g 3114	. . . . . . . 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 184	. . . . . 6 |- ( [ z / x ] w e. A <-> w e. [_ z / x ]_ A )
19	10, 14, 18	3bitr4g 223	. . . . 5 |- ( [_ y / x ]_ A = [_ z / x ]_ A -> ( [ y / x ] w e. A <-> [ z / x ] w e. A ) )
20	19	2alimi 1479	. . . 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 2009	. . . 4 |- ( F/ x w e. A <-> A. y A. z ( [ y / x ] w e. A <-> [ z / x ] w e. A ) )
22	20, 21	sylibr 134	. . 3 |- ( A. y A. z [_ y / x ]_ A = [_ z / x ]_ A -> F/ x w e. A )
23	9, 22	nfcd 2343	. 2 |- ( A. y A. z [_ y / x ]_ A = [_ z / x ]_ A -> F/_ x A )
24	8, 23	impbii 126	1 |- ( F/_ x A <-> A. y A. z [_ y / x ]_ A = [_ z / x ]_ A )

Syntax hints:   <-> wb 105   A.wal 1371   = wceq 1373   F/wnf 1483   [wsb 1785   e. wcel 2176   F/_wnfc 2335   _Vcvv 2772   [.wsbc 2998   [_csb 3093

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-sbc 2999  df-csb 3094

This theorem is referenced by:  eusvnf  4501