Theorem sbccog 1942

Description: A composition law for class substitution.

Assertion

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

Distinct variable group:   ph,y

Proof of Theorem sbccog

Step	Hyp	Ref	Expression
1		dfsbcq 1933	. 2 |- (z = A -> ([z / y][y / x]ph <-> [A / y][y / x]ph))
2		dfsbcq 1933	. 2 |- (z = A -> ([z / x]ph <-> [A / x]ph))
3		ax-17 968	. . 3 |- (ph -> A.yph)
4	3	sbco2 1250	. 2 |- ([z / y][y / x]ph <-> [z / x]ph)
5	1, 2, 4	vtoclbg 1839	1 |- (A e. B -> ([A / y][y / x]ph <-> [A / x]ph))

Syntax hints:   -> wi 3   <-> wb 146   e. wcel 955  [wsbc 1166

This theorem is referenced by:  elrabsf 1953  sbcel1gv 1970  sbcel2gv 1971  sbcgf 1976  sbccomg 1979  sbcralt 1980  sbcralgf 1982  csbcog 1997  sbcbrg 2652

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803  df-sbc 1932

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