Theorem sbequ12a 1921

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 1919	. 2 |- (x = y -> (ph <-> [y / x]ph))
2		sbequ12 1919	. . 3 |- (y = x -> (ph <-> [x / y]ph))
3	2	equcoms 1681	. 2 |- (x = y -> (ph <-> [x / y]ph))
4	1, 3	bitr3d 246	1 |- (x = y -> ([y / x]ph <-> [x / y]ph))

This proof depends on syntax axioms:   -> wi 4   <-> wb 176  [wsb 1648

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746

This proof depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-sb 1649

This theorem is used by:  sbco3  2088