Theorem sbimi 1652

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 13	. . 3 |- ((x = y -> ph) -> (x = y -> ps))
3	1	anim2i 552	. . . 4 |- ((x = y /\ ph) -> (x = y /\ ps))
4	3	eximi 1576	. . 3 |- (E.x(x = y /\ ph) -> E.x(x = y /\ ps))
5	2, 4	anim12i 549	. 2 |- (((x = y -> ph) /\ E.x(x = y /\ ph)) -> ((x = y -> ps) /\ E.x(x = y /\ ps)))
6		df-sb 1649	. 2 |- ([y / x]ph <-> ((x = y -> ph) /\ E.x(x = y /\ ph)))
7		df-sb 1649	. 2 |- ([y / x]ps <-> ((x = y -> ps) /\ E.x(x = y /\ ps)))
8	5, 6, 7	3imtr4i 257	1 |- ([y / x]ph -> [y / x]ps)

Syntax hints:   -> wi 4   /\ wa 358  E.wex 1541  [wsb 1648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-sb 1649

This theorem is referenced by:  sbbii  1653  sb6f  2039  hbsb3  2043  sbi2  2064  sbco  2083  sbidm  2085  sbal1  2126  sbal  2127