Theorem bj-dvdemo1 31461

Description: Remove dependency on ax-13 2101 from dvdemo1 4648 (this removal is noteworthy since dvdemo1 4648 and dvdemo2 4649 illustrate the phenomenon of bundling). (Contributed by BJ, 16-Jul-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-dvdemo1	|- E. x ( x = y -> z e. x )

Distinct variable group:   x, y

Proof of Theorem bj-dvdemo1

Step	Hyp	Ref	Expression
1		bj-dtru 31456	. . 3 |- -. A. x x = y
2		exnal 1709	. . 3 |- ( E. x -. x = y <-> -. A. x x = y )
3	1, 2	mpbir 214	. 2 |- E. x -. x = y
4		pm2.21 112	. 2 |- ( -. x = y -> ( x = y -> z e. x ) )
5	3, 4	eximii 1719	1 |- E. x ( x = y -> z e. x )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1452   E.wex 1673

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-nul 4547  ax-pow 4594

This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1457  df-ex 1674  df-nf 1678

This theorem is referenced by: (None)