Theorem sbequ2 1882

Description: An equality theorem for substitution. (Contributed by NM, 16-May-1993.) (Proof shortened by Wolf Lammen, 25-Feb-2018.)

Assertion

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

Proof of Theorem sbequ2

Step	Hyp	Ref	Expression
1		df-sb 1881	. . 3 |- ( [ y / x ] ph <-> ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) )
2	1	simplbi 476	. 2 |- ( [ y / x ] ph -> ( x = y -> ph ) )
3	2	com12 32	1 |- ( x = y -> ( [ y / x ] ph -> ph ) )

Syntax hints:   -> wi 4   /\ wa 384   E.wex 1704   [wsb 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

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

This theorem is referenced by:  stdpc7  1958  sbequ12  2111  dfsb2  2373  sbequi  2375  sbi1  2392  bj-mo3OLD  32832  2pm13.193  38768  2pm13.193VD  39139