Theorem sbequ1 1724

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

Assertion

Ref	Expression
sbequ1	|- ( x = y -> ( ph -> [ y / x ] ph ) )

Proof of Theorem sbequ1

Step	Hyp	Ref	Expression
1		pm3.4 329	. . 3 |- ( ( x = y /\ ph ) -> ( x = y -> ph ) )
2		19.8a 1552	. . 3 |- ( ( x = y /\ ph ) -> E. x ( x = y /\ ph ) )
3		df-sb 1719	. . 3 |- ( [ y / x ] ph <-> ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) )
4	1, 2, 3	sylanbrc 411	. 2 |- ( ( x = y /\ ph ) -> [ y / x ] ph )
5	4	ex 114	1 |- ( x = y -> ( ph -> [ y / x ] ph ) )

Syntax hints:   -> wi 4   /\ wa 103   E.wex 1451   [wsb 1718

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-4 1470

This theorem depends on definitions:  df-bi 116  df-sb 1719

This theorem is referenced by:  sbequ12  1727  sbequi  1793  sb6rf  1807  mo2n  2003  bj-bdfindes  12839  bj-findes  12871