Theorem sbid2 1251

Description: An identity law for substitution.

Hypothesis

Ref	Expression
sbid2.1	|- (ph -> A.xph)

Assertion

Ref	Expression
sbid2	|- ([y / x][x / y]ph <-> ph)

Proof of Theorem sbid2

Step	Hyp	Ref	Expression
1		sbco 1250	. 2 |- ([y / x][x / y]ph <-> [y / x]ph)
2		sbid2.1	. . 3 |- (ph -> A.xph)
3	2	sbf 1184	. 2 |- ([y / x]ph <-> ph)
4	1, 3	bitr 173	1 |- ([y / x][x / y]ph <-> ph)

Syntax hints:   -> wi 3   <-> wb 146  A.wal 952  [wsbc 1168

This theorem is referenced by:  sbco2 1253  sb5rf 1257  sb8 1259  sbid2v 1341

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-11o 1216

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170

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