Theorem bj-alsb 32625

Description: If a proposition is true for all instances, then it is true for any specific one. Uses only ax-1--5. Compare stdpc4 2353 which uses auxiliary axioms. (Contributed by BJ, 22-Dec-2020.)

Assertion

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

Proof of Theorem bj-alsb

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ala1 1741	. . . 4 |- ( A. x ph -> A. x ( x = y -> ph ) )
2	1	a1d 25	. . 3 |- ( A. x ph -> ( y = t -> A. x ( x = y -> ph ) ) )
3	2	alrimiv 1855	. 2 |- ( A. x ph -> A. y ( y = t -> A. x ( x = y -> ph ) ) )
4		df-ssb 32620	. 2 |- ([ t/ x]b ph <-> A. y ( y = t -> A. x ( x = y -> ph ) ) )
5	3, 4	sylibr 224	1 |- ( A. x 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

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

This theorem is referenced by:  bj-ssbft  32642