Theorem dveeq1-o 34220

Description: Quantifier introduction when one pair of variables is distinct. Version of dveeq1 2300 using ax-c11 . (Contributed by NM, 2-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
dveeq1-o	|- ( -. A. x x = y -> ( y = z -> A. x y = z ) )

Distinct variable group:   x, z

Proof of Theorem dveeq1-o

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-5 1839	. 2 |- ( w = z -> A. x w = z )
2		ax-5 1839	. 2 |- ( y = z -> A. w y = z )
3		equequ1 1952	. 2 |- ( w = y -> ( w = z <-> y = z ) )
4	1, 2, 3	dvelimf-o 34214	1 |- ( -. A. x x = y -> ( y = z -> A. x y = z ) )

Syntax hints:   -. wn 3   -> 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  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-c5 34168  ax-c4 34169  ax-c7 34170  ax-c10 34171  ax-c11 34172  ax-c9 34175

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

This theorem is referenced by:  ax12inda2ALT  34231