Theorem sblim 1957

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 1953	. 2 |- ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> [ y / x ] ps ) )
2		sblim.1	. . . 4 |- F/ x ps
3	2	sbf 1777	. . 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 1460   [wsb 1762

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763

This theorem is referenced by:  sbnf2  1981  sbmo  2085