Theorem sbimi 1687

Description: Infer substitution into antecedent and consequent of an implication. (Contributed by NM, 25-Jun-1998.)

Hypothesis

Ref	Expression
sbimi.1	|- ( ph -> ps )

Assertion

Ref	Expression
sbimi	|- ( [ y / x ] ph -> [ y / x ] ps )

Proof of Theorem sbimi

Step	Hyp	Ref	Expression
1		sbimi.1	. . . 4 |- ( ph -> ps )
2	1	imim2i 12	. . 3 |- ( ( x = y -> ph ) -> ( x = y -> ps ) )
3	1	anim2i 334	. . . 4 |- ( ( x = y /\ ph ) -> ( x = y /\ ps ) )
4	3	eximi 1531	. . 3 |- ( E. x ( x = y /\ ph ) -> E. x ( x = y /\ ps ) )
5	2, 4	anim12i 331	. 2 |- ( ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) -> ( ( x = y -> ps ) /\ E. x ( x = y /\ ps ) ) )
6		df-sb 1686	. 2 |- ( [ y / x ] ph <-> ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) )
7		df-sb 1686	. 2 |- ( [ y / x ] ps <-> ( ( x = y -> ps ) /\ E. x ( x = y /\ ps ) ) )
8	5, 6, 7	3imtr4i 199	1 |- ( [ y / x ] ph -> [ y / x ] ps )

Syntax hints:   -> wi 4   /\ wa 102   E.wex 1421   [wsb 1685

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-sb 1686

This theorem is referenced by:  sbbii  1688  sb6f  1724  hbsb3  1729  sbidm  1772  sbco  1883  sbcocom  1885  elsb3  1893  elsb4  1894  sbalyz  1916  hbsb4t  1930  moimv  2007  oprcl  3594  peano1  4335  peano2  4336