Theorem bj-ssbim 32621

Description: Distribute substitution over implication, closed form. Specialization of implication. Uses only ax-1--5. Compare spsbim 2394. (Contributed by BJ, 22-Dec-2020.)

Assertion

Ref	Expression
bj-ssbim	|- ( A. x ( ph -> ps ) -> ([ t/ x]b ph -> [ t/ x]b ps ) )

Proof of Theorem bj-ssbim

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		imim2 58	. . . . 5 |- ( ( ph -> ps ) -> ( ( x = y -> ph ) -> ( x = y -> ps ) ) )
2	1	al2imi 1743	. . . 4 |- ( A. x ( ph -> ps ) -> ( A. x ( x = y -> ph ) -> A. x ( x = y -> ps ) ) )
3	2	imim2d 57	. . 3 |- ( A. x ( ph -> ps ) -> ( ( y = t -> A. x ( x = y -> ph ) ) -> ( y = t -> A. x ( x = y -> ps ) ) ) )
4	3	alimdv 1845	. 2 |- ( A. x ( ph -> ps ) -> ( A. y ( y = t -> A. x ( x = y -> ph ) ) -> A. y ( y = t -> A. x ( x = y -> ps ) ) ) )
5		df-ssb 32620	. 2 |- ([ t/ x]b ph <-> A. y ( y = t -> A. x ( x = y -> ph ) ) )
6		df-ssb 32620	. 2 |- ([ t/ x]b ps <-> A. y ( y = t -> A. x ( x = y -> ps ) ) )
7	4, 5, 6	3imtr4g 285	1 |- ( A. x ( ph -> ps ) -> ([ t/ x]b ph -> [ t/ x]b ps ) )

Syntax hints:   -> wi 4   A.wal 1481  [wssb 32619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839

This theorem depends on definitions:  df-bi 197  df-ssb 32620

This theorem is referenced by:  bj-ssbbi  32622  bj-ssbimi  32623