Theorem sb8e 1257

Description: Substitution of variable in existential quantifier.

Hypothesis

Ref	Expression
sb8e.1	|- (ph -> A.yph)

Assertion

Ref	Expression
sb8e	|- (E.xph <-> E.y[y / x]ph)

Proof of Theorem sb8e

Step	Hyp	Ref	Expression
1		sb8e.1	. . . . . 6 |- (ph -> A.yph)
2	1	hbn 1001	. . . . 5 |- (-. ph -> A.y -. ph)
3	2	sb8 1256	. . . 4 |- (A.x -. ph <-> A.y[y / x] -. ph)
4		sbn 1226	. . . . 5 |- ([y / x] -. ph <-> -. [y / x]ph)
5	4	albii 996	. . . 4 |- (A.y[y / x] -. ph <-> A.y -. [y / x]ph)
6	3, 5	bitr 173	. . 3 |- (A.x -. ph <-> A.y -. [y / x]ph)
7	6	negbii 187	. 2 |- (-. A.x -. ph <-> -. A.y -. [y / x]ph)
8		df-ex 978	. 2 |- (E.xph <-> -. A.x -. ph)
9		df-ex 978	. 2 |- (E.y[y / x]ph <-> -. A.y -. [y / x]ph)
10	7, 8, 9	3bitr4 183	1 |- (E.xph <-> E.y[y / x]ph)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146  A.wal 951  E.wex 977  [wsbc 1166

This theorem is referenced by:  exsb 1345

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-10 963  ax-11 964  ax-12 965  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-11o 1213

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

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