Theorem sbcco 2899

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 2886	. 2 |- ( [. A / y ]. [. y / x ]. ph -> A e. _V )
2		sbcex 2886	. 2 |- ( [. A / x ]. ph -> A e. _V )
3		dfsbcq 2880	. . 3 |- ( z = A -> ( [. z / y ]. [. y / x ]. ph <-> [. A / y ]. [. y / x ]. ph ) )
4		dfsbcq 2880	. . 3 |- ( z = A -> ( [. z / x ]. ph <-> [. A / x ]. ph ) )
5		sbsbc 2882	. . . . . 6 |- ( [ y / x ] ph <-> [. y / x ]. ph )
6	5	sbbii 1721	. . . . 5 |- ( [ z / y ] [ y / x ] ph <-> [ z / y ] [. y / x ]. ph )
7		nfv 1491	. . . . . 6 |- F/ y ph
8	7	sbco2 1914	. . . . 5 |- ( [ z / y ] [ y / x ] ph <-> [ z / x ] ph )
9		sbsbc 2882	. . . . 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 2882	. . . 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 2718	. 2 |- ( A e. _V -> ( [. A / y ]. [. y / x ]. ph <-> [. A / x ]. ph ) )
14	1, 2, 13	pm5.21nii 676	1 |- ( [. A / y ]. [. y / x ]. ph <-> [. A / x ]. ph )

Syntax hints:   <-> wb 104   e. wcel 1463   [wsb 1718   _Vcvv 2657   [.wsbc 2878

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-sbc 2879

This theorem is referenced by:  sbc7  2904  sbccom  2952  sbcralt  2953  sbcrext  2954  csbco  2980