Theorem wl-euequ1f 33356

Description: euequ1 2476 proved with a distinctor. (Contributed by Wolf Lammen, 23-Sep-2020.)

Assertion

Ref	Expression
wl-euequ1f	|- ( -. A. x x = y -> E! x x = y )

Proof of Theorem wl-euequ1f

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax6ev 1890	. . 3 |- E. z z = y
2		nfv 1843	. . . 4 |- F/ z -. A. x x = y
3		nfnae 2318	. . . . 5 |- F/ x -. A. x x = y
4		nfeqf2 2297	. . . . 5 |- ( -. A. x x = y -> F/ x z = y )
5		equequ2 1953	. . . . . . 7 |- ( y = z -> ( x = y <-> x = z ) )
6	5	equcoms 1947	. . . . . 6 |- ( z = y -> ( x = y <-> x = z ) )
7	6	a1i 11	. . . . 5 |- ( -. A. x x = y -> ( z = y -> ( x = y <-> x = z ) ) )
8	3, 4, 7	alrimdd 2083	. . . 4 |- ( -. A. x x = y -> ( z = y -> A. x ( x = y <-> x = z ) ) )
9	2, 8	eximd 2085	. . 3 |- ( -. A. x x = y -> ( E. z z = y -> E. z A. x ( x = y <-> x = z ) ) )
10	1, 9	mpi 20	. 2 |- ( -. A. x x = y -> E. z A. x ( x = y <-> x = z ) )
11		df-eu 2474	. 2 |- ( E! x x = y <-> E. z A. x ( x = y <-> x = z ) )
12	10, 11	sylibr 224	1 |- ( -. A. x x = y -> E! x x = y )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   A.wal 1481   E.wex 1704   E!weu 2470

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-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-eu 2474

This theorem is referenced by: (None)