Theorem bj-ssbbi 32622

Description: Biconditional property for substitution, closed form. Specialization of biconditional. Uses only ax-1--5. Compare spsbbi 2402. (Contributed by BJ, 22-Dec-2020.)

Assertion

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

Proof of Theorem bj-ssbbi

Step	Hyp	Ref	Expression
1		biimp 205	. . . 4 |- ( ( ph <-> ps ) -> ( ph -> ps ) )
2	1	alimi 1739	. . 3 |- ( A. x ( ph <-> ps ) -> A. x ( ph -> ps ) )
3		bj-ssbim 32621	. . 3 |- ( A. x ( ph -> ps ) -> ([ t/ x]b ph -> [ t/ x]b ps ) )
4	2, 3	syl 17	. 2 |- ( A. x ( ph <-> ps ) -> ([ t/ x]b ph -> [ t/ x]b ps ) )
5		biimpr 210	. . . 4 |- ( ( ph <-> ps ) -> ( ps -> ph ) )
6	5	alimi 1739	. . 3 |- ( A. x ( ph <-> ps ) -> A. x ( ps -> ph ) )
7		bj-ssbim 32621	. . 3 |- ( A. x ( ps -> ph ) -> ([ t/ x]b ps -> [ t/ x]b ph ) )
8	6, 7	syl 17	. 2 |- ( A. x ( ph <-> ps ) -> ([ t/ x]b ps -> [ t/ x]b ph ) )
9	4, 8	impbid 202	1 |- ( A. x ( ph <-> ps ) -> ([ t/ x]b ph <-> [ t/ x]b ps ) )

Syntax hints:   -> wi 4   <-> wb 196   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-ssbbii  32624