Theorem sbimi 1810

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 342	. . . 4 |- ( ( x = y /\ ph ) -> ( x = y /\ ps ) )
4	3	eximi 1646	. . 3 |- ( E. x ( x = y /\ ph ) -> E. x ( x = y /\ ps ) )
5	2, 4	anim12i 338	. 2 |- ( ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) -> ( ( x = y -> ps ) /\ E. x ( x = y /\ ps ) ) )
6		df-sb 1809	. 2 |- ( [ y / x ] ph <-> ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) )
7		df-sb 1809	. 2 |- ( [ y / x ] ps <-> ( ( x = y -> ps ) /\ E. x ( x = y /\ ps ) ) )
8	5, 6, 7	3imtr4i 201	1 |- ( [ y / x ] ph -> [ y / x ] ps )

Syntax hints:   -> wi 4   /\ wa 104   E.wex 1538   [wsb 1808

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-ial 1580

This theorem depends on definitions:  df-bi 117  df-sb 1809

This theorem is referenced by:  sbbii  1811  sb6f  1849  hbsb3  1854  sbidm  1897  sbco  2019  sbcocom  2021  sbalyz  2050  hbsb4t  2064  moimv  2144  elsb1  2207  elsb2  2208  oprcl  3880  peano1  4685  peano2  4686