Theorem sbequ12a 1947

Description: An equality theorem for substitution. (Contributed by NM, 2-Jun-1993.) (Proof shortened by Wolf Lammen, 23-Jun-2019.)

Assertion

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

Proof of Theorem sbequ12a

Step	Hyp	Ref	Expression
1		sbequ12r 1946	. 2 |- ( x = y -> ( [ x / y ] ph <-> ph ) )
2		sbequ12 1945	. 2 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
3	1, 2	bitr2d 254	1 |- ( x = y -> ( [ y / x ] ph <-> [ x / y ] ph ) )

Syntax hints:   -> wi 4   <-> wb 184   [wsb 1702

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-12 1794

This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1588  df-sb 1703

This theorem is referenced by:  sbco3  2121  sbco3OLD  2122  sb9  2133