Theorem sbi2v 1813

Description: Reverse direction of sbimv 1814. (Contributed by Jim Kingdon, 18-Jan-2018.)

Assertion

Ref	Expression
sbi2v	|- ( ( [ y / x ] ph -> [ y / x ] ps ) -> [ y / x ] ( ph -> ps ) )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)   ps( x, y)

Proof of Theorem sbi2v

Step	Hyp	Ref	Expression
1		19.38 1606	. . 3 |- ( ( E. x ( x = y /\ ph ) -> A. x ( x = y -> ps ) ) -> A. x ( ( x = y /\ ph ) -> ( x = y -> ps ) ) )
2		pm3.3 257	. . . . 5 |- ( ( ( x = y /\ ph ) -> ( x = y -> ps ) ) -> ( x = y -> ( ph -> ( x = y -> ps ) ) ) )
3		pm2.04 81	. . . . 5 |- ( ( ph -> ( x = y -> ps ) ) -> ( x = y -> ( ph -> ps ) ) )
4	2, 3	syli 37	. . . 4 |- ( ( ( x = y /\ ph ) -> ( x = y -> ps ) ) -> ( x = y -> ( ph -> ps ) ) )
5	4	alimi 1384	. . 3 |- ( A. x ( ( x = y /\ ph ) -> ( x = y -> ps ) ) -> A. x ( x = y -> ( ph -> ps ) ) )
6	1, 5	syl 14	. 2 |- ( ( E. x ( x = y /\ ph ) -> A. x ( x = y -> ps ) ) -> A. x ( x = y -> ( ph -> ps ) ) )
7		sb5 1808	. . 3 |- ( [ y / x ] ph <-> E. x ( x = y /\ ph ) )
8		sb6 1807	. . 3 |- ( [ y / x ] ps <-> A. x ( x = y -> ps ) )
9	7, 8	imbi12i 237	. 2 |- ( ( [ y / x ] ph -> [ y / x ] ps ) <-> ( E. x ( x = y /\ ph ) -> A. x ( x = y -> ps ) ) )
10		sb6 1807	. 2 |- ( [ y / x ] ( ph -> ps ) <-> A. x ( x = y -> ( ph -> ps ) ) )
11	6, 9, 10	3imtr4i 199	1 |- ( ( [ y / x ] ph -> [ y / x ] ps ) -> [ y / x ] ( ph -> ps ) )

Syntax hints:   -> wi 4   /\ wa 102   A.wal 1282   E.wex 1421   [wsb 1685

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-sb 1686

This theorem is referenced by:  sbimv  1814