Theorem sbcexg 1975

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 1943	. 2 |- (z = A -> ([z / y]E.xph <-> [A / y]E.xph))
2		dfsbcq 1943	. . 3 |- (z = A -> ([z / y]ph <-> [A / y]ph))
3	2	exbidv 1279	. 2 |- (z = A -> (E.x[z / y]ph <-> E.x[A / y]ph))
4		sbex 1348	. 2 |- ([z / y]E.xph <-> E.x[z / y]ph)
5	1, 3, 4	vtoclbg 1848	1 |- (A e. B -> ([A / y]E.xph <-> E.x[A / y]ph))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 956   e. wcel 958  E.wex 980  [wsbc 1170

This theorem is referenced by:  sbcabel 1996  sbcel12g 2011  csbopabg 2678

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-sbc 1942

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