Theorem sbcco 3020

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 3007	. 2 |- ( [. A / y ]. [. y / x ]. ph -> A e. _V )
2		sbcex 3007	. 2 |- ( [. A / x ]. ph -> A e. _V )
3		dfsbcq 3000	. . 3 |- ( z = A -> ( [. z / y ]. [. y / x ]. ph <-> [. A / y ]. [. y / x ]. ph ) )
4		dfsbcq 3000	. . 3 |- ( z = A -> ( [. z / x ]. ph <-> [. A / x ]. ph ) )
5		sbsbc 3002	. . . . . 6 |- ( [ y / x ] ph <-> [. y / x ]. ph )
6	5	sbbii 1788	. . . . 5 |- ( [ z / y ] [ y / x ] ph <-> [ z / y ] [. y / x ]. ph )
7		nfv 1551	. . . . . 6 |- F/ y ph
8	7	sbco2 1993	. . . . 5 |- ( [ z / y ] [ y / x ] ph <-> [ z / x ] ph )
9		sbsbc 3002	. . . . 5 |- ( [ z / y ] [. y / x ]. ph <-> [. z / y ]. [. y / x ]. ph )
10	6, 8, 9	3bitr3ri 211	. . . 4 |- ( [. z / y ]. [. y / x ]. ph <-> [ z / x ] ph )
11		sbsbc 3002	. . . 4 |- ( [ z / x ] ph <-> [. z / x ]. ph )
12	10, 11	bitri 184	. . 3 |- ( [. z / y ]. [. y / x ]. ph <-> [. z / x ]. ph )
13	3, 4, 12	vtoclbg 2834	. 2 |- ( A e. _V -> ( [. A / y ]. [. y / x ]. ph <-> [. A / x ]. ph ) )
14	1, 2, 13	pm5.21nii 706	1 |- ( [. A / y ]. [. y / x ]. ph <-> [. A / x ]. ph )

Syntax hints:   <-> wb 105   [wsb 1785   e. wcel 2176   _Vcvv 2772   [.wsbc 2998

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-sbc 2999

This theorem is referenced by:  sbc7  3025  sbccom  3074  sbcralt  3075  sbcrext  3076  csbco  3103