Theorem sbcco 2972

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 2959	. 2 |- ( [. A / y ]. [. y / x ]. ph -> A e. _V )
2		sbcex 2959	. 2 |- ( [. A / x ]. ph -> A e. _V )
3		dfsbcq 2953	. . 3 |- ( z = A -> ( [. z / y ]. [. y / x ]. ph <-> [. A / y ]. [. y / x ]. ph ) )
4		dfsbcq 2953	. . 3 |- ( z = A -> ( [. z / x ]. ph <-> [. A / x ]. ph ) )
5		sbsbc 2955	. . . . . 6 |- ( [ y / x ] ph <-> [. y / x ]. ph )
6	5	sbbii 1753	. . . . 5 |- ( [ z / y ] [ y / x ] ph <-> [ z / y ] [. y / x ]. ph )
7		nfv 1516	. . . . . 6 |- F/ y ph
8	7	sbco2 1953	. . . . 5 |- ( [ z / y ] [ y / x ] ph <-> [ z / x ] ph )
9		sbsbc 2955	. . . . 5 |- ( [ z / y ] [. y / x ]. ph <-> [. z / y ]. [. y / x ]. ph )
10	6, 8, 9	3bitr3ri 210	. . . 4 |- ( [. z / y ]. [. y / x ]. ph <-> [ z / x ] ph )
11		sbsbc 2955	. . . 4 |- ( [ z / x ] ph <-> [. z / x ]. ph )
12	10, 11	bitri 183	. . 3 |- ( [. z / y ]. [. y / x ]. ph <-> [. z / x ]. ph )
13	3, 4, 12	vtoclbg 2787	. 2 |- ( A e. _V -> ( [. A / y ]. [. y / x ]. ph <-> [. A / x ]. ph ) )
14	1, 2, 13	pm5.21nii 694	1 |- ( [. A / y ]. [. y / x ]. ph <-> [. A / x ]. ph )

Syntax hints:   <-> wb 104   [wsb 1750   e. wcel 2136   _Vcvv 2726   [.wsbc 2951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-sbc 2952

This theorem is referenced by:  sbc7  2977  sbccom  3026  sbcralt  3027  sbcrext  3028  csbco  3055