Theorem sbimi 854

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	syl3 18	. . 3 |- ((x = y -> ph) -> (x = y -> ps))
3	1	anim2i 270	. . . 4 |- ((x = y /\ ph) -> (x = y /\ ps))
4	3	19.22i 723	. . 3 |- (E.x(x = y /\ ph) -> E.x(x = y /\ ps))
5	2, 4	anim12i 268	. 2 |- (((x = y -> ph) /\ E.x(x = y /\ ph)) -> ((x = y -> ps) /\ E.x(x = y /\ ps)))
6		df-sb 853	. 2 |- ([y / x]ph <-> ((x = y -> ph) /\ E.x(x = y /\ ph)))
7		df-sb 853	. 2 |- ([y / x]ps <-> ((x = y -> ps) /\ E.x(x = y /\ ps)))
8	5, 6, 7	3imtr4 192	1 |- ([y / x]ph -> [y / x]ps)

Syntax hints:   -> wi 2   /\ wa 196  E.wex 678   = weq 797  [wsb 852

This theorem is referenced by:  bisb 855  sbi2 885  sbco 910  sbal1 996  sbal 997  tfinds2 2405

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853

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