Theorem bj-ax12ssb 32635

Description: The axiom bj-ax12 32634 expressed using substitution. (Contributed by BJ, 26-Dec-2020.)

Assertion

Ref	Expression
bj-ax12ssb	|- [ t/ x]b ( ph -> [ t/ x]b ph )

Distinct variable group:   x, t

Allowed substitution hints:   ph( x, t)

Proof of Theorem bj-ax12ssb

Step	Hyp	Ref	Expression
1		bj-ax12 32634	. . 3 |- A. x ( x = t -> ( ph -> A. x ( x = t -> ph ) ) )
2		bj-ssb1 32633	. . . . . 6 |- ([ t/ x]b ph <-> A. x ( x = t -> ph ) )
3	2	imbi2i 326	. . . . 5 |- ( ( ph -> [ t/ x]b ph ) <-> ( ph -> A. x ( x = t -> ph ) ) )
4	3	imbi2i 326	. . . 4 |- ( ( x = t -> ( ph -> [ t/ x]b ph ) ) <-> ( x = t -> ( ph -> A. x ( x = t -> ph ) ) ) )
5	4	albii 1747	. . 3 |- ( A. x ( x = t -> ( ph -> [ t/ x]b ph ) ) <-> A. x ( x = t -> ( ph -> A. x ( x = t -> ph ) ) ) )
6	1, 5	mpbir 221	. 2 |- A. x ( x = t -> ( ph -> [ t/ x]b ph ) )
7		bj-ssb1 32633	. 2 |- ([ t/ x]b ( ph -> [ t/ x]b ph ) <-> A. x ( x = t -> ( ph -> [ t/ x]b ph ) ) )
8	6, 7	mpbir 221	1 |- [ t/ x]b ( ph -> [ t/ x]b ph )

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  ax-6 1888  ax-7 1935  ax-11 2034  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-ssb 32620

This theorem is referenced by: (None)