Theorem sbccsbg 2012

Description: Substitution into a wff expressed in terms of substitution into a class.

Assertion

Ref	Expression
sbccsbg	|- (A e. B -> ([A / x]ph <-> y e. [_A / x]_{y | ph}))

Distinct variable group:   x,y

Proof of Theorem sbccsbg

Step	Hyp	Ref	Expression
1		abid 1458	. . 3 |- (y e. {y | ph} <-> ph)
2	1	sbcbii 1968	. 2 |- (A e. B -> ([A / x]y e. {y | ph} <-> [A / x]ph))
3		sbcel2g 2005	. 2 |- (A e. B -> ([A / x]y e. {y | ph} <-> y e. [_A / x]_{y | ph}))
4	2, 3	bitr3d 528	1 |- (A e. B -> ([A / x]ph <-> y e. [_A / x]_{y | ph}))

Syntax hints:   -> wi 3   <-> wb 146   e. wcel 955  [wsbc 1166  {cab 1456  [_csb 1991

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-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803  df-sbc 1932  df-csb 1992

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