Theorem sbequ12a 1784

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

Assertion

Ref	Expression
sbequ12a	|- ( x = y -> ( [ y / x ] ph <-> [ x / y ] ph ) )

Proof of Theorem sbequ12a

Step	Hyp	Ref	Expression
1		sbequ12 1782	. 2 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
2		sbequ12 1782	. . 3 |- ( y = x -> ( ph <-> [ x / y ] ph ) )
3	2	equcoms 1719	. 2 |- ( x = y -> ( ph <-> [ x / y ] ph ) )
4	1, 3	bitr3d 190	1 |- ( x = y -> ( [ y / x ] ph <-> [ x / y ] ph ) )

Syntax hints:   -> wi 4   <-> wb 105   [wsb 1773

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-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-4 1521  ax-17 1537  ax-i9 1541

This theorem depends on definitions:  df-bi 117  df-sb 1774

This theorem is referenced by: (None)