Theorem sbimi 1764

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 1600	. . 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 1763	. 2 |- ( [ y / x ] ph <-> ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) )
7		df-sb 1763	. 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 1492   [wsb 1762

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 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-ial 1534

This theorem depends on definitions:  df-bi 117  df-sb 1763

This theorem is referenced by:  sbbii  1765  sb6f  1803  hbsb3  1808  sbidm  1851  sbco  1968  sbcocom  1970  sbalyz  1999  hbsb4t  2013  moimv  2092  elsb1  2155  elsb2  2156  oprcl  3804  peano1  4595  peano2  4596