Theorem sbccog 2467

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 2455	. 2 |- (z = A -> ([z / y][y / x]ph <-> [A / y][y / x]ph))
2		dfsbcq 2455	. 2 |- (z = A -> ([z / x]ph <-> [A / x]ph))
3		ax-17 1317	. . 3 |- (ph -> A.yph)
4	3	sbco2 1629	. 2 |- ([z / y][y / x]ph <-> [z / x]ph)
5	1, 2, 4	vtoclbg 2347	1 |- (A e. B -> ([A / y][y / x]ph <-> [A / x]ph))

Syntax hints:   -> wi 3   <-> wb 163   e. wcel 1300  [wsbc 1534

This theorem is referenced by:  sbc7g 2476  elrabsf 2486  sbcel1gvOLD 2511  sbcel2gvOLD 2513  sbcgfOLD 2521  sbccomg 2526  sbcralt 2527  sbcralgf 2529  csbcog 2547  sbcbrgOLD 3391  bnj610 12564  bnj972 12858  bnj974 12859  bnj1377 13095  bnj1468 13145

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-sbc 2454

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