Theorem aevlem 1981

Description: Lemma for aev 1983 and axc16g 2134. Change free and bound variables. Instance of aev 1983. (Contributed by NM, 22-Jul-2015.) (Proof shortened by Wolf Lammen, 17-Feb-2018.) Remove dependency on ax-13 2246, along an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) (Revised by BJ, 29-Mar-2021.)

Assertion

Ref	Expression
aevlem	|- ( A. x x = y -> A. z z = t )

Distinct variable groups:   x, y   z, t

Proof of Theorem aevlem

Dummy variable u is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbvaev 1979	. 2 |- ( A. x x = y -> A. u u = y )
2		aevlem0 1980	. 2 |- ( A. u u = y -> A. x x = u )
3		cbvaev 1979	. 2 |- ( A. x x = u -> A. t t = u )
4		aevlem0 1980	. 2 |- ( A. t t = u -> A. z z = t )
5	1, 2, 3, 4	4syl 19	1 |- ( A. x x = y -> A. z z = t )

Syntax hints:   -> wi 4   A.wal 1481

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:  aeveq  1982  aev  1983  hbaevg  1984  axc16g  2134  axc11vOLD  2141  axc16gOLD  2161  aevOLD  2162  aevALTOLD  2321  bj-axc16g16  32674  bj-axc11nv  32745