Theorem sbimi 1173

Description: Infer substitution into antecedent and consequent of an implication.

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 17	. . 3 |- ((x = y -> ph) -> (x = y -> ps))
3	1	anim2i 335	. . . 4 |- ((x = y /\ ph) -> (x = y /\ ps))
4	3	19.22i 1040	. . 3 |- (E.x(x = y /\ ph) -> E.x(x = y /\ ps))
5	2, 4	anim12i 333	. 2 |- (((x = y -> ph) /\ E.x(x = y /\ ph)) -> ((x = y -> ps) /\ E.x(x = y /\ ps)))
6		df-sb 1172	. 2 |- ([y / x]ph <-> ((x = y -> ph) /\ E.x(x = y /\ ph)))
7		df-sb 1172	. 2 |- ([y / x]ps <-> ((x = y -> ps) /\ E.x(x = y /\ ps)))
8	5, 6, 7	3imtr4 219	1 |- ([y / x]ph -> [y / x]ps)

Syntax hints:   -> wi 3   /\ wa 223  E.wex 980  [wsbc 1170

This theorem is referenced by:  sbbii 1174  sb6f 1201  hbsb3 1206  sbi2 1233  sbco 1252  equsb3lem 1329  elsb3 1331  sbal1 1346  sbal 1347  tfinds2 3165  csbfsum 7027

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 963  ax-4 973  ax-5o 975

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172

Copyright terms: Public domain