Theorem sbid2 1253

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 1252	. 2 |- ([y / x][x / y]ph <-> [y / x]ph)
2		sbid2.1	. . 3 |- (ph -> A.xph)
3	2	sbf 1186	. 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 954  [wsbc 1170

This theorem is referenced by:  sbco2 1255  sb5rf 1259  sb8 1261  sbid2v 1343

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-11o 1218

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172

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