Theorem sbnfc2 4007

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 3203	. . . . 5 |- y e. _V
2		csbtt 3544	. . . . 5 |- ( ( y e. _V /\ F/_ x A ) -> [_ y / x ]_ A = A )
3	1, 2	mpan 706	. . . 4 |- ( F/_ x A -> [_ y / x ]_ A = A )
4		vex 3203	. . . . 5 |- z e. _V
5		csbtt 3544	. . . . 5 |- ( ( z e. _V /\ F/_ x A ) -> [_ z / x ]_ A = A )
6	4, 5	mpan 706	. . . 4 |- ( F/_ x A -> [_ z / x ]_ A = A )
7	3, 6	eqtr4d 2659	. . 3 |- ( F/_ x A -> [_ y / x ]_ A = [_ z / x ]_ A )
8	7	alrimivv 1856	. 2 |- ( F/_ x A -> A. y A. z [_ y / x ]_ A = [_ z / x ]_ A )
9		nfv 1843	. . 3 |- F/ w A. y A. z [_ y / x ]_ A = [_ z / x ]_ A
10		eleq2 2690	. . . . . 6 |- ( [_ y / x ]_ A = [_ z / x ]_ A -> ( w e. [_ y / x ]_ A <-> w e. [_ z / x ]_ A ) )
11		sbsbc 3439	. . . . . . 7 |- ( [ y / x ] w e. A <-> [. y / x ]. w e. A )
12		sbcel2 3989	. . . . . . 7 |- ( [. y / x ]. w e. A <-> w e. [_ y / x ]_ A )
13	11, 12	bitri 264	. . . . . 6 |- ( [ y / x ] w e. A <-> w e. [_ y / x ]_ A )
14		sbsbc 3439	. . . . . . 7 |- ( [ z / x ] w e. A <-> [. z / x ]. w e. A )
15		sbcel2 3989	. . . . . . 7 |- ( [. z / x ]. w e. A <-> w e. [_ z / x ]_ A )
16	14, 15	bitri 264	. . . . . 6 |- ( [ z / x ] w e. A <-> w e. [_ z / x ]_ A )
17	10, 13, 16	3bitr4g 303	. . . . 5 |- ( [_ y / x ]_ A = [_ z / x ]_ A -> ( [ y / x ] w e. A <-> [ z / x ] w e. A ) )
18	17	2alimi 1740	. . . 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 ) )
19		sbnf2 2439	. . . 4 |- ( F/ x w e. A <-> A. y A. z ( [ y / x ] w e. A <-> [ z / x ] w e. A ) )
20	18, 19	sylibr 224	. . 3 |- ( A. y A. z [_ y / x ]_ A = [_ z / x ]_ A -> F/ x w e. A )
21	9, 20	nfcd 2759	. 2 |- ( A. y A. z [_ y / x ]_ A = [_ z / x ]_ A -> F/_ x A )
22	8, 21	impbii 199	1 |- ( F/_ x A <-> A. y A. z [_ y / x ]_ A = [_ z / x ]_ A )

Syntax hints:   <-> wb 196   A.wal 1481   = wceq 1483   F/wnf 1708   [wsb 1880   e. wcel 1990   F/_wnfc 2751   _Vcvv 3200   [.wsbc 3435   [_csb 3533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-nul 3916

This theorem is referenced by:  eusvnf  4861