Theorem sbimi 1752

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 340	. . . 4 |- ( ( x = y /\ ph ) -> ( x = y /\ ps ) )
4	3	eximi 1588	. . 3 |- ( E. x ( x = y /\ ph ) -> E. x ( x = y /\ ps ) )
5	2, 4	anim12i 336	. 2 |- ( ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) -> ( ( x = y -> ps ) /\ E. x ( x = y /\ ps ) ) )
6		df-sb 1751	. 2 |- ( [ y / x ] ph <-> ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) )
7		df-sb 1751	. 2 |- ( [ y / x ] ps <-> ( ( x = y -> ps ) /\ E. x ( x = y /\ ps ) ) )
8	5, 6, 7	3imtr4i 200	1 |- ( [ y / x ] ph -> [ y / x ] ps )

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  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-ial 1522

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

This theorem is referenced by:  sbbii  1753  sb6f  1791  hbsb3  1796  sbidm  1839  sbco  1956  sbcocom  1958  sbalyz  1987  hbsb4t  2001  moimv  2080  elsb1  2143  elsb2  2144  oprcl  3782  peano1  4571  peano2  4572