Theorem sbnfc2 3197

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

Assertion

Ref	Expression
sbnfc2	|- ( F/_xA <-> A.yA.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 2863	. . . . 5 |- y e. _V
2		csbtt 3149	. . . . 5 |- ((y e. _V /\ F/_xA) -> [_y / x]_A = A)
3	1, 2	mpan 651	. . . 4 |- ( F/_xA -> [_y / x]_A = A)
4		vex 2863	. . . . 5 |- z e. _V
5		csbtt 3149	. . . . 5 |- ((z e. _V /\ F/_xA) -> [_z / x]_A = A)
6	4, 5	mpan 651	. . . 4 |- ( F/_xA -> [_z / x]_A = A)
7	3, 6	eqtr4d 2388	. . 3 |- ( F/_xA -> [_y / x]_A = [_z / x]_A)
8	7	alrimivv 1632	. 2 |- ( F/_xA -> A.yA.z[_y / x]_A = [_z / x]_A)
9		nfv 1619	. . 3 |- F/wA.yA.z[_y / x]_A = [_z / x]_A
10		eleq2 2414	. . . . . 6 |- ([_y / x]_A = [_z / x]_A -> (w e. [_y / x]_A <-> w e. [_z / x]_A))
11		sbsbc 3051	. . . . . . 7 |- ([y / x]w e. A <-> [.y / x ].w e. A)
12		sbcel2g 3158	. . . . . . . 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 240	. . . . . 6 |- ([y / x]w e. A <-> w e. [_y / x]_A)
15		sbsbc 3051	. . . . . . 7 |- ([z / x]w e. A <-> [.z / x ].w e. A)
16		sbcel2g 3158	. . . . . . . 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 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 1560	. . . 4 |- (A.yA.z[_y / x]_A = [_z / x]_A -> A.yA.z([y / x]w e. A <-> [z / x]w e. A))
21		sbnf2 2108	. . . 4 |- ( F/x w e. A <-> A.yA.z([y / x]w e. A <-> [z / x]w e. A))
22	20, 21	sylibr 203	. . 3 |- (A.yA.z[_y / x]_A = [_z / x]_A -> F/x w e. A)
23	9, 22	nfcd 2485	. 2 |- (A.yA.z[_y / x]_A = [_z / x]_A -> F/_xA)
24	8, 23	impbii 180	1 |- ( F/_xA <-> A.yA.z[_y / x]_A = [_z / x]_A)

Syntax hints:   <-> wb 176  A.wal 1540   F/wnf 1544   = wceq 1642  [wsb 1648   e. wcel 1710   F/_wnfc 2477  _Vcvv 2860   [.wsbc 3047  [_csb 3137

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-sbc 3048  df-csb 3138

This theorem is referenced by: (None)