Theorem sbequ2 1757

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 1751	. 2 |- ( [ y / x ] ph <-> ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) )
2		simpl 108	. . 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	syl5bi 151	1 |- ( x = y -> ( [ y / x ] ph -> ph ) )

Syntax hints:   -> wi 4   /\ wa 103   E.wex 1480   [wsb 1750

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105

This theorem depends on definitions:  df-bi 116  df-sb 1751

This theorem is referenced by:  stdpc7  1758  sbequ12  1759  sbequi  1827  mo23  2055  mopick  2092