Theorem dveeq2 2024

Description: Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.) (Revised by NM, 20-Jul-2015.)

Assertion

Ref	Expression
dveeq2	|- ( -. A. x x = y -> ( z = y -> A. x z = y ) )

Distinct variable group:   x, z

Proof of Theorem dveeq2

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		equequ2 1694	. 2 |- ( w = y -> ( z = w <-> z = y ) )
2	1	dvelimv 2020	1 |- ( -. A. x x = y -> ( z = y -> A. x z = y ) )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1546

This theorem is referenced by:  ax11v2  2025  ax11v2OLD  2026  sbal1  2184  copsexg  4410  axpowndlem3  8438

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946

This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551