Theorem ax9 2003

Description: Proof of ax-9 1999 from ax9v1 2001 and ax9v2 2002, proving sufficiency of the conjunction of the latter two weakened versions of ax9v 2000, which is itself a weakened version of ax-9 1999. (Contributed by BJ, 7-Dec-2020.) (Proof shortened by Wolf Lammen, 11-Apr-2021.)

Assertion

Ref	Expression
ax9	|- ( x = y -> ( z e. x -> z e. y ) )

Proof of Theorem ax9

Dummy variable t is distinct from all other variables.

Step	Hyp	Ref	Expression
1		equvinv 1959	. 2 |- ( x = y <-> E. t ( t = x /\ t = y ) )
2		ax9v2 2002	. . . . 5 |- ( x = t -> ( z e. x -> z e. t ) )
3	2	equcoms 1947	. . . 4 |- ( t = x -> ( z e. x -> z e. t ) )
4		ax9v1 2001	. . . 4 |- ( t = y -> ( z e. t -> z e. y ) )
5	3, 4	sylan9 689	. . 3 |- ( ( t = x /\ t = y ) -> ( z e. x -> z e. y ) )
6	5	exlimiv 1858	. 2 |- ( E. t ( t = x /\ t = y ) -> ( z e. x -> z e. y ) )
7	1, 6	sylbi 207	1 |- ( x = y -> ( z e. x -> z e. y ) )

Syntax hints:   -> wi 4   /\ wa 384   E.wex 1704

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-9 1999

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

This theorem is referenced by:  elequ2  2004  el  4847  dtru  4857  fv3  6206  elirrv  8504  bj-ax89  32667  bj-el  32796  bj-dtru  32797  axc11next  38607