Theorem sbequ12a 1656

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 1654	. 2 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
2		sbequ12 1654	. . 3 |- ( y = x -> ( ph <-> [ x / y ] ph ) )
3	2	equcoms 1594	. 2 |- ( x = y -> ( ph <-> [ x / y ] ph ) )
4	1, 3	bitr3d 179	1 |- ( x = y -> ( [ y / x ] ph <-> [ x / y ] ph ) )

Syntax hints:   -> wi 4   <-> wb 98   [wsb 1645

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423

This theorem depends on definitions:  df-bi 110  df-sb 1646

This theorem is referenced by: (None)