Theorem sbid2h 1849

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 1810	. 2 |- ( [ y / x ] [ x / y ] ph <-> [ y / x ] ph )
3	1	sbh 1776	. 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 1351   [wsb 1762

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 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534

This theorem depends on definitions:  df-bi 117  df-sb 1763

This theorem is referenced by:  sbid2  1850  sb5rf  1852  sb6rf  1853  sbid2v  1996