Theorem bj-sbex 32626

Description: If a proposition is true for a specific instance, then there exists an instance such that it is true for it. Uses only ax-1--6. Compare spsbe 1884 which, due to the specific form of df-sb 1881, uses fewer axioms. (Contributed by BJ, 22-Dec-2020.)

Assertion

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

Proof of Theorem bj-sbex

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ssb 32620	. . 3 |- ([ t/ x]b ph <-> A. y ( y = t -> A. x ( x = y -> ph ) ) )
2		ax6ev 1890	. . . 4 |- E. y y = t
3		exim 1761	. . . 4 |- ( A. y ( y = t -> A. x ( x = y -> ph ) ) -> ( E. y y = t -> E. y A. x ( x = y -> ph ) ) )
4	2, 3	mpi 20	. . 3 |- ( A. y ( y = t -> A. x ( x = y -> ph ) ) -> E. y A. x ( x = y -> ph ) )
5	1, 4	sylbi 207	. 2 |- ([ t/ x]b ph -> E. y A. x ( x = y -> ph ) )
6		exim 1761	. . 3 |- ( A. x ( x = y -> ph ) -> ( E. x x = y -> E. x ph ) )
7	6	eximi 1762	. 2 |- ( E. y A. x ( x = y -> ph ) -> E. y ( E. x x = y -> E. x ph ) )
8		ax6ev 1890	. . . 4 |- E. x x = y
9		pm2.27 42	. . . 4 |- ( E. x x = y -> ( ( E. x x = y -> E. x ph ) -> E. x ph ) )
10	8, 9	ax-mp 5	. . 3 |- ( ( E. x x = y -> E. x ph ) -> E. x ph )
11	10	exlimiv 1858	. 2 |- ( E. y ( E. x x = y -> E. x ph ) -> E. x ph )
12	5, 7, 11	3syl 18	1 |- ([ t/ x]b ph -> E. x 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

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

This theorem is referenced by:  bj-ssbft  32642