Theorem sbrim 2016

Description: Substitution with a variable not free in antecedent affects only the consequent. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
sbrim.1	|- ( ph -> A. x ph )

Assertion

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

Proof of Theorem sbrim

Step	Hyp	Ref	Expression
1		sbim 2013	. 2 |- ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> [ y / x ] ps ) )
2		sbrim.1	. . . 4 |- ( ph -> A. x ph )
3	2	sbh 1829	. . 3 |- ( [ y / x ] ph <-> ph )
4	3	imbi1i 238	. 2 |- ( ( [ y / x ] ph -> [ y / x ] ps ) <-> ( ph -> [ y / x ] ps ) )
5	1, 4	bitri 184	1 |- ( [ y / x ] ( ph -> ps ) <-> ( ph -> [ y / x ] ps ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1400   [wsb 1815

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816

This theorem is referenced by:  sbco2d  2026  sbco2vd  2027  hbsbd  2042