Theorem sb8 918

Description: Substitution of variable in universal quantifier.

Hypothesis

Ref	Expression
sb8.1	|- (ph -> A.yph)

Assertion

Ref	Expression
sb8	|- (A.xph <-> A.y[y / x]ph)

Proof of Theorem sb8

Step	Hyp	Ref	Expression
1		sb8.1	. . . 4 |- (ph -> A.yph)
2	1	hbal 700	. . 3 |- (A.xph -> A.yA.xph)
3		stdpc4 869	. . 3 |- (A.xph -> [y / x]ph)
4	2, 3	19.21ai 740	. 2 |- (A.xph -> A.y[y / x]ph)
5	1	hbsb3 875	. . . 4 |- ([y / x]ph -> A.x[y / x]ph)
6	5	hbal 700	. . 3 |- (A.y[y / x]ph -> A.xA.y[y / x]ph)
7		stdpc4 869	. . . 4 |- (A.y[y / x]ph -> [x / y][y / x]ph)
8	1	sbid2 911	. . . 4 |- ([x / y][y / x]ph <-> ph)
9	7, 8	sylib 173	. . 3 |- (A.y[y / x]ph -> ph)
10	6, 9	19.21ai 740	. 2 |- (A.y[y / x]ph -> A.xph)
11	4, 10	impbi 139	1 |- (A.xph <-> A.y[y / x]ph)

Syntax hints:   -> wi 2   <-> wb 127  A.wal 672  [wsb 852

This theorem is referenced by:  sb8e 919  sb8eu 1017

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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