Theorem sbid2h 1842

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 1803	. 2 |- ( [ y / x ] [ x / y ] ph <-> [ y / x ] ph )
3	1	sbh 1769	. 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 1346   [wsb 1755

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 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-sb 1756

This theorem is referenced by:  sbid2  1843  sb5rf  1845  sb6rf  1846  sbid2v  1989