Theorem sbequ2 1783

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

Assertion

Ref	Expression
sbequ2	|- ( x = y -> ( [ y / x ] ph -> ph ) )

Proof of Theorem sbequ2

Step	Hyp	Ref	Expression
1		df-sb 1777	. 2 |- ( [ y / x ] ph <-> ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) )
2		simpl 109	. . 3 |- ( ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) -> ( x = y -> ph ) )
3	2	com12 30	. 2 |- ( x = y -> ( ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) -> ph ) )
4	1, 3	biimtrid 152	1 |- ( x = y -> ( [ y / x ] ph -> ph ) )

Syntax hints:   -> wi 4   /\ wa 104   E.wex 1506   [wsb 1776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106

This theorem depends on definitions:  df-bi 117  df-sb 1777

This theorem is referenced by:  stdpc7  1784  sbequ12  1785  sbequi  1853  mo23  2086  mopick  2123