Theorem spsbbi 1866

Description: Specialization of biconditional. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 21-Jan-2018.)

Assertion

Ref	Expression
spsbbi	|- ( A. x ( ph <-> ps ) -> ( [ y / x ] ph <-> [ y / x ] ps ) )

Proof of Theorem spsbbi

Step	Hyp	Ref	Expression
1		spsbim 1865	. . 3 |- ( A. x ( ph -> ps ) -> ( [ y / x ] ph -> [ y / x ] ps ) )
2		spsbim 1865	. . 3 |- ( A. x ( ps -> ph ) -> ( [ y / x ] ps -> [ y / x ] ph ) )
3	1, 2	anim12i 338	. 2 |- ( ( A. x ( ph -> ps ) /\ A. x ( ps -> ph ) ) -> ( ( [ y / x ] ph -> [ y / x ] ps ) /\ ( [ y / x ] ps -> [ y / x ] ph ) ) )
4		albiim 1509	. 2 |- ( A. x ( ph <-> ps ) <-> ( A. x ( ph -> ps ) /\ A. x ( ps -> ph ) ) )
5		dfbi2 388	. 2 |- ( ( [ y / x ] ph <-> [ y / x ] ps ) <-> ( ( [ y / x ] ph -> [ y / x ] ps ) /\ ( [ y / x ] ps -> [ y / x ] ph ) ) )
6	3, 4, 5	3imtr4i 201	1 |- ( A. x ( ph <-> ps ) -> ( [ y / x ] ph <-> [ y / x ] ps ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1370   [wsb 1784

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-5 1469  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-4 1532  ax-ial 1556

This theorem depends on definitions:  df-bi 117  df-sb 1785

This theorem is referenced by:  sbbidh  1867  sbbid  1868  sbbidv  1907  relelfvdm  5607