Theorem sbid2h 1873

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 1834	. 2 |- ( [ y / x ] [ x / y ] ph <-> [ y / x ] ph )
3	1	sbh 1800	. 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 1371   [wsb 1786

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 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558

This theorem depends on definitions:  df-bi 117  df-sb 1787

This theorem is referenced by:  sbid2  1874  sb5rf  1876  sb6rf  1877  sbid2v  2025