Theorem equviniva 1960

Description: A modified version of the forward implication of equvinv 1959 adapted to common usage. (Contributed by Wolf Lammen, 8-Sep-2018.)

Assertion

Ref	Expression
equviniva	|- ( x = y -> E. z ( x = z /\ y = z ) )

Distinct variable groups:   x, z   y, z

Proof of Theorem equviniva

Step	Hyp	Ref	Expression
1		ax6evr 1942	. 2 |- E. z y = z
2		equtr 1948	. . . 4 |- ( x = y -> ( y = z -> x = z ) )
3	2	ancrd 577	. . 3 |- ( x = y -> ( y = z -> ( x = z /\ y = z ) ) )
4	3	eximdv 1846	. 2 |- ( x = y -> ( E. z y = z -> E. z ( x = z /\ y = z ) ) )
5	1, 4	mpi 20	1 |- ( x = y -> E. z ( x = z /\ y = z ) )

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

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

This theorem is referenced by:  equvinvOLD  1962  ax13lem1  2248  nfeqf  2301  bj-ssbequ2  32643  wl-ax13lem1  33287