Theorem sbi1v 1903

Description: Forward direction of sbimv 1905. (Contributed by Jim Kingdon, 25-Dec-2017.)

Assertion

Ref	Expression
sbi1v	|- ( [ y / x ] ( ph -> ps ) -> ( [ y / x ] ph -> [ y / x ] ps ) )

Distinct variable group:   x, y

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

Proof of Theorem sbi1v

Step	Hyp	Ref	Expression
1		sb6 1898	. 2 |- ( [ y / x ] ph <-> A. x ( x = y -> ph ) )
2		sb6 1898	. . 3 |- ( [ y / x ] ( ph -> ps ) <-> A. x ( x = y -> ( ph -> ps ) ) )
3		ax-2 7	. . . . 5 |- ( ( x = y -> ( ph -> ps ) ) -> ( ( x = y -> ph ) -> ( x = y -> ps ) ) )
4	3	al2imi 1469	. . . 4 |- ( A. x ( x = y -> ( ph -> ps ) ) -> ( A. x ( x = y -> ph ) -> A. x ( x = y -> ps ) ) )
5		sb2 1778	. . . 4 |- ( A. x ( x = y -> ps ) -> [ y / x ] ps )
6	4, 5	syl6 33	. . 3 |- ( A. x ( x = y -> ( ph -> ps ) ) -> ( A. x ( x = y -> ph ) -> [ y / x ] ps ) )
7	2, 6	sylbi 121	. 2 |- ( [ y / x ] ( ph -> ps ) -> ( A. x ( x = y -> ph ) -> [ y / x ] ps ) )
8	1, 7	biimtrid 152	1 |- ( [ y / x ] ( ph -> ps ) -> ( [ y / x ] ph -> [ y / x ] ps ) )

Syntax hints:   -> wi 4   A.wal 1362   [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

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

This theorem is referenced by:  sbimv  1905