Theorem sb8 1265

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 1009	. . 3 |- (A.xph -> A.yA.xph)
3		stdpc4 1189	. . 3 |- (A.xph -> [y / x]ph)
4	2, 3	19.21ai 1002	. 2 |- (A.xph -> A.y[y / x]ph)
5	1	hbsb3 1210	. . . 4 |- ([y / x]ph -> A.x[y / x]ph)
6	5	hbal 1009	. . 3 |- (A.y[y / x]ph -> A.xA.y[y / x]ph)
7		stdpc4 1189	. . . 4 |- (A.y[y / x]ph -> [x / y][y / x]ph)
8	1	sbid2 1257	. . . 4 |- ([x / y][y / x]ph <-> ph)
9	7, 8	sylib 198	. . 3 |- (A.y[y / x]ph -> ph)
10	6, 9	19.21ai 1002	. 2 |- (A.y[y / x]ph -> A.xph)
11	4, 10	impbii 157	1 |- (A.xph <-> A.y[y / x]ph)

Syntax hints:   -> wi 3   <-> wb 146  A.wal 958  [wsbc 1174

This theorem is referenced by:  sb8e 1266  sb8eu 1394

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-10 970  ax-11 971  ax-12 972  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1127  ax-10o 1144  ax-11o 1222

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 985  df-sb 1176

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