Theorem sbcco 1448

Description: A composition law for class substitution.

Assertion

Ref	Expression
sbcco	|- (A e. B -> ([A / y][y / x]ph <-> [A / x]ph))

Distinct variable group(s):   ph,y

Proof of Theorem sbcco

Step	Hyp	Ref	Expression
1		dfsbcq 1442	. 2 |- (z = A -> ([z / y][y / x]ph <-> [A / y][y / x]ph))
2		dfsbcq 1442	. 2 |- (z = A -> ([z / x]ph <-> [A / x]ph))
3		ax-17 925	. . 3 |- (ph -> A.yph)
4	3	sbco2 913	. 2 |- ([z / y][y / x]ph <-> [z / x]ph)
5	1, 2, 4	vtoclbg 1384	1 |- (A e. B -> ([A / y][y / x]ph <-> [A / x]ph))

Syntax hints:   -> wi 2   <-> wb 127  [wsb 852   e. wcel 1092  [wsbc 1440

This theorem is referenced by:  elrabsf 1456  sbcgf 1469

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-sbc 1441

metamath.org