Theorem sblim 1872

Description: Substitution with a variable not free in consequent affects only the antecedent. (Contributed by NM, 14-Nov-2013.) (Revised by Mario Carneiro, 4-Oct-2016.)

Hypothesis

Ref	Expression
sblim.1	|- F/ x ps

Assertion

Ref	Expression
sblim	|- ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> ps ) )

Proof of Theorem sblim

Step	Hyp	Ref	Expression
1		sbim 1868	. 2 |- ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> [ y / x ] ps ) )
2		sblim.1	. . . 4 |- F/ x ps
3	2	sbf 1700	. . 3 |- ( [ y / x ] ps <-> ps )
4	3	imbi2i 224	. 2 |- ( ( [ y / x ] ph -> [ y / x ] ps ) <-> ( [ y / x ] ph -> ps ) )
5	1, 4	bitri 182	1 |- ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> ps ) )

Syntax hints:   -> wi 4   <-> wb 103   F/wnf 1389   [wsb 1685

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686

This theorem is referenced by:  sbnf2  1898  sbmo  2000