Theorem wl-19.2reqv 33310

Description: Under the assumption -. x = y the reverse direction of 19.2 1892 is provable from Tarski's FOL and ax13v 2247 only. Note that in conjunction with 19.2 1892 in fact ( -. x = y -> ( A. x z = y <-> E. x z = y ) ) holds. (Contributed by Wolf Lammen, 17-Apr-2021.)

Assertion

Ref	Expression
wl-19.2reqv	|- ( -. x = y -> ( E. x z = y -> A. x z = y ) )

Distinct variable group:   x, z

Proof of Theorem wl-19.2reqv

Step	Hyp	Ref	Expression
1		ax13lem2 2296	. 2 |- ( -. x = y -> ( E. x z = y -> z = y ) )
2		ax13lem1 2248	. 2 |- ( -. x = y -> ( z = y -> A. x z = y ) )
3	1, 2	syld 47	1 |- ( -. x = y -> ( E. x z = y -> A. x z = y ) )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1481   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-13 2246

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

This theorem is referenced by: (None)