Theorem sbcexg 1971

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

Assertion

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

Distinct variable groups:   x,A   x,y

Proof of Theorem sbcexg

Step	Hyp	Ref	Expression
1		dfsbcq 1939	. 2 |- (z = A -> ([z / y]E.xph <-> [A / y]E.xph))
2		dfsbcq 1939	. . 3 |- (z = A -> ([z / y]ph <-> [A / y]ph))
3	2	exbidv 1277	. 2 |- (z = A -> (E.x[z / y]ph <-> E.x[A / y]ph))
4		sbex 1346	. 2 |- ([z / y]E.xph <-> E.x[z / y]ph)
5	1, 3, 4	vtoclbg 1844	1 |- (A e. B -> ([A / y]E.xph <-> E.x[A / y]ph))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 954   e. wcel 956  E.wex 978  [wsbc 1168

This theorem is referenced by:  sbcabel 1992  sbcel12g 2007  csbopabg 2673

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808  df-sbc 1938

Copyright terms: Public domain