Theorem sbequ12a 2113

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 2112	. 2 |- ( x = y -> ( [ x / y ] ph <-> ph ) )
2		sbequ12 2111	. 2 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
3	1, 2	bitr2d 269	1 |- ( x = y -> ( [ y / x ] ph <-> [ x / y ] ph ) )

Syntax hints:   -> wi 4   <-> wb 196   [wsb 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-sb 1881

This theorem is referenced by:  sbco3  2417  sb9  2426