Theorem sbcco 2837

Description: A composition law for class substitution. (Contributed by NM, 26-Sep-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)

Assertion

Ref	Expression
sbcco	|- ( [. A / y ]. [. y / x ]. ph <-> [. A / x ]. ph )

Distinct variable group:   ph, y

Allowed substitution hints:   ph( x)   A( x, y)

Proof of Theorem sbcco

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbcex 2824	. 2 |- ( [. A / y ]. [. y / x ]. ph -> A e. _V )
2		sbcex 2824	. 2 |- ( [. A / x ]. ph -> A e. _V )
3		dfsbcq 2818	. . 3 |- ( z = A -> ( [. z / y ]. [. y / x ]. ph <-> [. A / y ]. [. y / x ]. ph ) )
4		dfsbcq 2818	. . 3 |- ( z = A -> ( [. z / x ]. ph <-> [. A / x ]. ph ) )
5		sbsbc 2820	. . . . . 6 |- ( [ y / x ] ph <-> [. y / x ]. ph )
6	5	sbbii 1689	. . . . 5 |- ( [ z / y ] [ y / x ] ph <-> [ z / y ] [. y / x ]. ph )
7		nfv 1462	. . . . . 6 |- F/ y ph
8	7	sbco2 1881	. . . . 5 |- ( [ z / y ] [ y / x ] ph <-> [ z / x ] ph )
9		sbsbc 2820	. . . . 5 |- ( [ z / y ] [. y / x ]. ph <-> [. z / y ]. [. y / x ]. ph )
10	6, 8, 9	3bitr3ri 209	. . . 4 |- ( [. z / y ]. [. y / x ]. ph <-> [ z / x ] ph )
11		sbsbc 2820	. . . 4 |- ( [ z / x ] ph <-> [. z / x ]. ph )
12	10, 11	bitri 182	. . 3 |- ( [. z / y ]. [. y / x ]. ph <-> [. z / x ]. ph )
13	3, 4, 12	vtoclbg 2660	. 2 |- ( A e. _V -> ( [. A / y ]. [. y / x ]. ph <-> [. A / x ]. ph ) )
14	1, 2, 13	pm5.21nii 653	1 |- ( [. A / y ]. [. y / x ]. ph <-> [. A / x ]. ph )

Syntax hints:   <-> wb 103   e. wcel 1434   [wsb 1686   _Vcvv 2602   [.wsbc 2816

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-sbc 2817

This theorem is referenced by:  sbc7  2842  sbccom  2890  sbcralt  2891  sbcrext  2892  csbco  2918