Theorem sbid2h 1822

Description: An identity law for substitution. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
sbid2h.1	|- ( ph -> A. x ph )

Assertion

Ref	Expression
sbid2h	|- ( [ y / x ] [ x / y ] ph <-> ph )

Proof of Theorem sbid2h

Step	Hyp	Ref	Expression
1		sbid2h.1	. . 3 |- ( ph -> A. x ph )
2	1	sbcof2 1783	. 2 |- ( [ y / x ] [ x / y ] ph <-> [ y / x ] ph )
3	1	sbh 1750	. 2 |- ( [ y / x ] ph <-> ph )
4	2, 3	bitri 183	1 |- ( [ y / x ] [ x / y ] ph <-> ph )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1330   [wsb 1736

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515

This theorem depends on definitions:  df-bi 116  df-sb 1737

This theorem is referenced by:  sbid2  1823  sb5rf  1825  sb6rf  1826  sbid2v  1972