Theorem sb8 1256

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 1002	. . 3 |- (A.xph -> A.yA.xph)
3		stdpc4 1181	. . 3 |- (A.xph -> [y / x]ph)
4	2, 3	19.21ai 995	. 2 |- (A.xph -> A.y[y / x]ph)
5	1	hbsb3 1202	. . . 4 |- ([y / x]ph -> A.x[y / x]ph)
6	5	hbal 1002	. . 3 |- (A.y[y / x]ph -> A.xA.y[y / x]ph)
7		stdpc4 1181	. . . 4 |- (A.y[y / x]ph -> [x / y][y / x]ph)
8	1	sbid2 1248	. . . 4 |- ([x / y][y / x]ph <-> ph)
9	7, 8	sylib 198	. . 3 |- (A.y[y / x]ph -> ph)
10	6, 9	19.21ai 995	. 2 |- (A.y[y / x]ph -> A.xph)
11	4, 10	impbi 157	1 |- (A.xph <-> A.y[y / x]ph)

Syntax hints:   -> wi 3   <-> wb 146  A.wal 951  [wsbc 1166

This theorem is referenced by:  sb8e 1257  sb8eu 1383

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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