Theorem bj-ssb1a 32632

Description: One direction of a simplified definition of substitution in case of disjoint variables. See bj-ssb1 32633 for the biconditional, which requires ax-11 2034. (Contributed by BJ, 22-Dec-2020.)

Assertion

Ref	Expression
bj-ssb1a	|- ( A. x ( x = t -> ph ) -> [ t/ x]b ph )

Distinct variable group:   x, t

Allowed substitution hints:   ph( x, t)

Proof of Theorem bj-ssb1a

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-1 6	. . . . . 6 |- ( ( x = t -> ph ) -> ( E. y y = t -> ( x = t -> ph ) ) )
2		19.23v 1902	. . . . . 6 |- ( A. y ( y = t -> ( x = t -> ph ) ) <-> ( E. y y = t -> ( x = t -> ph ) ) )
3	1, 2	sylibr 224	. . . . 5 |- ( ( x = t -> ph ) -> A. y ( y = t -> ( x = t -> ph ) ) )
4		equequ2 1953	. . . . . . . 8 |- ( y = t -> ( x = y <-> x = t ) )
5	4	imbi1d 331	. . . . . . 7 |- ( y = t -> ( ( x = y -> ph ) <-> ( x = t -> ph ) ) )
6	5	pm5.74i 260	. . . . . 6 |- ( ( y = t -> ( x = y -> ph ) ) <-> ( y = t -> ( x = t -> ph ) ) )
7	6	albii 1747	. . . . 5 |- ( A. y ( y = t -> ( x = y -> ph ) ) <-> A. y ( y = t -> ( x = t -> ph ) ) )
8	3, 7	sylibr 224	. . . 4 |- ( ( x = t -> ph ) -> A. y ( y = t -> ( x = y -> ph ) ) )
9	8	alimi 1739	. . 3 |- ( A. x ( x = t -> ph ) -> A. x A. y ( y = t -> ( x = y -> ph ) ) )
10		bj-ssblem2 32631	. . 3 |- ( A. x A. y ( y = t -> ( x = y -> ph ) ) -> A. y A. x ( y = t -> ( x = y -> ph ) ) )
11		stdpc5v 1867	. . . 4 |- ( A. x ( y = t -> ( x = y -> ph ) ) -> ( y = t -> A. x ( x = y -> ph ) ) )
12	11	alimi 1739	. . 3 |- ( A. y A. x ( y = t -> ( x = y -> ph ) ) -> A. y ( y = t -> A. x ( x = y -> ph ) ) )
13	9, 10, 12	3syl 18	. 2 |- ( A. x ( x = t -> ph ) -> A. y ( y = t -> A. x ( x = y -> ph ) ) )
14		df-ssb 32620	. 2 |- ([ t/ x]b ph <-> A. y ( y = t -> A. x ( x = y -> ph ) ) )
15	13, 14	sylibr 224	1 |- ( A. x ( x = t -> ph ) -> [ t/ x]b ph )

Syntax hints:   -> wi 4   A.wal 1481   E.wex 1704  [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

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

This theorem is referenced by: (None)