Theorem sbex 1343

Description: Move existential quantifier in and out of substitution.

Assertion

Ref	Expression
sbex	|- ([z / y]E.xph <-> E.x[z / y]ph)

Distinct variable groups:   x,y   x,z

Proof of Theorem sbex

Step	Hyp	Ref	Expression
1		sbn 1226	. . 3 |- ([z / y] -. A.x -. ph <-> -. [z / y]A.x -. ph)
2		sbal 1342	. . . . 5 |- ([z / y]A.x -. ph <-> A.x[z / y] -. ph)
3		sbn 1226	. . . . . 6 |- ([z / y] -. ph <-> -. [z / y]ph)
4	3	albii 996	. . . . 5 |- (A.x[z / y] -. ph <-> A.x -. [z / y]ph)
5	2, 4	bitr 173	. . . 4 |- ([z / y]A.x -. ph <-> A.x -. [z / y]ph)
6	5	negbii 187	. . 3 |- (-. [z / y]A.x -. ph <-> -. A.x -. [z / y]ph)
7	1, 6	bitr 173	. 2 |- ([z / y] -. A.x -. ph <-> -. A.x -. [z / y]ph)
8		df-ex 978	. . 3 |- (E.xph <-> -. A.x -. ph)
9	8	sbbii 1170	. 2 |- ([z / y]E.xph <-> [z / y] -. A.x -. ph)
10		df-ex 978	. 2 |- (E.x[z / y]ph <-> -. A.x -. [z / y]ph)
11	7, 9, 10	3bitr4 183	1 |- ([z / y]E.xph <-> E.x[z / y]ph)

Syntax hints:  -. wn 2   <-> wb 146  A.wal 951  E.wex 977  [wsbc 1166

This theorem is referenced by:  sbabel 1576  sbcexg 1965

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-12 965  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168

Copyright terms: Public domain