Theorem sbimi 1886

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 16	. . 3 |- ( ( x = y -> ph ) -> ( x = y -> ps ) )
3	1	anim2i 593	. . . 4 |- ( ( x = y /\ ph ) -> ( x = y /\ ps ) )
4	3	eximi 1762	. . 3 |- ( E. x ( x = y /\ ph ) -> E. x ( x = y /\ ps ) )
5	2, 4	anim12i 590	. 2 |- ( ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) -> ( ( x = y -> ps ) /\ E. x ( x = y /\ ps ) ) )
6		df-sb 1881	. 2 |- ( [ y / x ] ph <-> ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) )
7		df-sb 1881	. 2 |- ( [ y / x ] ps <-> ( ( x = y -> ps ) /\ E. x ( x = y /\ ps ) ) )
8	5, 6, 7	3imtr4i 281	1 |- ( [ y / x ] ph -> [ y / x ] ps )

Syntax hints:   -> wi 4   /\ wa 384   E.wex 1704   [wsb 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-sb 1881

This theorem is referenced by:  sbbii  1887  hbsb3  2364  sb6f  2385  sbi2  2393  sbie  2408  2mo  2551  fmptdF  29456  funcnv4mpt  29470  disjdsct  29480  measiuns  30280  ballotlemodife  30559  bj-hbsb3v  32761  bj-sbidmOLD  32831  mptsnunlem  33185