Theorem sbcco 2836

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 2823	. 2 |- ( [. A / y ]. [. y / x ]. ph -> A e. _V )
2		sbcex 2823	. 2 |- ( [. A / x ]. ph -> A e. _V )
3		dfsbcq 2817	. . 3 |- ( z = A -> ( [. z / y ]. [. y / x ]. ph <-> [. A / y ]. [. y / x ]. ph ) )
4		dfsbcq 2817	. . 3 |- ( z = A -> ( [. z / x ]. ph <-> [. A / x ]. ph ) )
5		sbsbc 2819	. . . . . 6 |- ( [ y / x ] ph <-> [. y / x ]. ph )
6	5	sbbii 1688	. . . . 5 |- ( [ z / y ] [ y / x ] ph <-> [ z / y ] [. y / x ]. ph )
7		nfv 1461	. . . . . 6 |- F/ y ph
8	7	sbco2 1880	. . . . 5 |- ( [ z / y ] [ y / x ] ph <-> [ z / x ] ph )
9		sbsbc 2819	. . . . 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 2819	. . . 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 2659	. 2 |- ( A e. _V -> ( [. A / y ]. [. y / x ]. ph <-> [. A / x ]. ph ) )
14	1, 2, 13	pm5.21nii 652	1 |- ( [. A / y ]. [. y / x ]. ph <-> [. A / x ]. ph )

Syntax hints:   <-> wb 103   e. wcel 1433   [wsb 1685   _Vcvv 2601   [.wsbc 2815

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-sbc 2816

This theorem is referenced by:  sbc7  2841  sbccom  2889  sbcralt  2890  sbcrext  2891  csbco  2917