Theorem sbi2v 1904

Description: Reverse direction of sbimv 1905. (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 1687	. . 3 |- ( ( E. x ( x = y /\ ph ) -> A. x ( x = y -> ps ) ) -> A. x ( ( x = y /\ ph ) -> ( x = y -> ps ) ) )
2		pm3.3 261	. . . . 5 |- ( ( ( x = y /\ ph ) -> ( x = y -> ps ) ) -> ( x = y -> ( ph -> ( x = y -> ps ) ) ) )
3		pm2.04 82	. . . . 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 1466	. . 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 1899	. . 3 |- ( [ y / x ] ph <-> E. x ( x = y /\ ph ) )
8		sb6 1898	. . 3 |- ( [ y / x ] ps <-> A. x ( x = y -> ps ) )
9	7, 8	imbi12i 239	. 2 |- ( ( [ y / x ] ph -> [ y / x ] ps ) <-> ( E. x ( x = y /\ ph ) -> A. x ( x = y -> ps ) ) )
10		sb6 1898	. 2 |- ( [ y / x ] ( ph -> ps ) <-> A. x ( x = y -> ( ph -> ps ) ) )
11	6, 9, 10	3imtr4i 201	1 |- ( ( [ y / x ] ph -> [ y / x ] ps ) -> [ y / x ] ( ph -> ps ) )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1362   E.wex 1503   [wsb 1773

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546

This theorem depends on definitions:  df-bi 117  df-sb 1774

This theorem is referenced by:  sbimv  1905