Theorem sblim 1944

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 1940	. 2 |- ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> [ y / x ] ps ) )
2		sblim.1	. . . 4 |- F/ x ps
3	2	sbf 1764	. . 3 |- ( [ y / x ] ps <-> ps )
4	3	imbi2i 225	. 2 |- ( ( [ y / x ] ph -> [ y / x ] ps ) <-> ( [ y / x ] ph -> ps ) )
5	1, 4	bitri 183	1 |- ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> ps ) )

Syntax hints:   -> wi 4   <-> wb 104   F/wnf 1447   [wsb 1749

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522

This theorem depends on definitions:  df-bi 116  df-nf 1448  df-sb 1750

This theorem is referenced by:  sbnf2  1968  sbmo  2072