Theorem sb8e 919

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	hbne 699	. . . . 5 |- (-. ph -> A.y -. ph)
3	2	sb8 918	. . . 4 |- (A.x -. ph <-> A.y[y / x] -. ph)
4		sbn 882	. . . . 5 |- ([y / x] -. ph <-> -. [y / x]ph)
5	4	bial 695	. . . 4 |- (A.y[y / x] -. ph <-> A.y -. [y / x]ph)
6	3, 5	bitr 151	. . 3 |- (A.x -. ph <-> A.y -. [y / x]ph)
7	6	negbii 162	. 2 |- (-. A.x -. ph <-> -. A.y -. [y / x]ph)
8		df-ex 679	. 2 |- (E.xph <-> -. A.x -. ph)
9		df-ex 679	. 2 |- (E.y[y / x]ph <-> -. A.y -. [y / x]ph)
10	7, 8, 9	3bitr4 158	1 |- (E.xph <-> E.y[y / x]ph)

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127  A.wal 672  E.wex 678  [wsb 852

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853

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