Theorem sbcalg 1984

Description: Move universal quantifier in and out of class substitution.

Assertion

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

Distinct variable groups:   x,A   x,y

Proof of Theorem sbcalg

Step	Hyp	Ref	Expression
1		dfsbcq 1950	. 2 |- (z = A -> ([z / y]A.xph <-> [A / y]A.xph))
2		dfsbcq 1950	. . 3 |- (z = A -> ([z / y]ph <-> [A / y]ph))
3	2	albidv 1282	. 2 |- (z = A -> (A.x[z / y]ph <-> A.x[A / y]ph))
4		sbal 1351	. 2 |- ([z / y]A.xph <-> A.x[z / y]ph)
5	1, 3, 4	vtoclbg 1855	1 |- (A e. B -> ([A / y]A.xph <-> A.x[A / y]ph))

Syntax hints:   -> wi 3   <-> wb 146  A.wal 958   = wceq 960   e. wcel 962  [wsbc 1174

This theorem is referenced by:  sbcabel 2006  sbcel12g 2022  sbceqdig 2023

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-9 969  ax-10 970  ax-12 972  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1127  ax-10o 1144  ax-16 1214  ax-11o 1222  ax-ext 1464

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 985  df-sb 1176  df-clab 1470  df-cleq 1475  df-clel 1478  df-v 1819  df-sbc 1949

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