Theorem sblim 2010

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 2006	. 2 |- ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> [ y / x ] ps ) )
2		sblim.1	. . . 4 |- F/ x ps
3	2	sbf 1825	. . 3 |- ( [ y / x ] ps <-> ps )
4	3	imbi2i 226	. 2 |- ( ( [ y / x ] ph -> [ y / x ] ps ) <-> ( [ y / x ] ph -> ps ) )
5	1, 4	bitri 184	1 |- ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> ps ) )

Syntax hints:   -> wi 4   <-> wb 105   F/wnf 1509   [wsb 1810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811

This theorem is referenced by:  sbnf2  2034  sbmo  2139