Theorem sbcco 3069

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 3056	. 2 |- ( [.A / y ]. [.y / x ].ph -> A e. _V)
2		sbcex 3056	. 2 |- ( [.A / x ].ph -> A e. _V)
3		dfsbcq 3049	. . 3 |- (z = A -> ( [.z / y ]. [.y / x ].ph <-> [.A / y ]. [.y / x ].ph))
4		dfsbcq 3049	. . 3 |- (z = A -> ( [.z / x ].ph <-> [.A / x ].ph))
5		sbsbc 3051	. . . . . 6 |- ([y / x]ph <-> [.y / x ].ph)
6	5	sbbii 1653	. . . . 5 |- ([z / y][y / x]ph <-> [z / y] [.y / x ].ph)
7		nfv 1619	. . . . . 6 |- F/yph
8	7	sbco2 2086	. . . . 5 |- ([z / y][y / x]ph <-> [z / x]ph)
9		sbsbc 3051	. . . . 5 |- ([z / y] [.y / x ].ph <-> [.z / y ]. [.y / x ].ph)
10	6, 8, 9	3bitr3ri 267	. . . 4 |- ( [.z / y ]. [.y / x ].ph <-> [z / x]ph)
11		sbsbc 3051	. . . 4 |- ([z / x]ph <-> [.z / x ].ph)
12	10, 11	bitri 240	. . 3 |- ( [.z / y ]. [.y / x ].ph <-> [.z / x ].ph)
13	3, 4, 12	vtoclbg 2916	. 2 |- (A e. _V -> ( [.A / y ]. [.y / x ].ph <-> [.A / x ].ph))
14	1, 2, 13	pm5.21nii 342	1 |- ( [.A / y ]. [.y / x ].ph <-> [.A / x ].ph)

Syntax hints:   <-> wb 176  [wsb 1648   e. wcel 1710  _Vcvv 2860   [.wsbc 3047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-sbc 3048

This theorem is referenced by:  sbc7  3074  sbccom  3118  sbcralt  3119  csbco  3146