Theorem sbid2h 1895

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 1856	. 2 |- ( [ y / x ] [ x / y ] ph <-> [ y / x ] ph )
3	1	sbh 1822	. 2 |- ( [ y / x ] ph <-> ph )
4	2, 3	bitri 184	1 |- ( [ y / x ] [ x / y ] ph <-> ph )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1393   [wsb 1808

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580

This theorem depends on definitions:  df-bi 117  df-sb 1809

This theorem is referenced by:  sbid2  1896  sb5rf  1898  sb6rf  1899  sbid2v  2047