Theorem dveeq2-o 34218

Description: Quantifier introduction when one pair of variables is distinct. Version of dveeq2 2298 using ax-c15 34174. (Contributed by NM, 2-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Distinct variable group:   x, z

Proof of Theorem dveeq2-o

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-5 1839	. 2 |- ( z = w -> A. x z = w )
2		ax-5 1839	. 2 |- ( z = y -> A. w z = y )
3		equequ2 1953	. 2 |- ( w = y -> ( z = w <-> z = y ) )
4	1, 2, 3	dvelimf-o 34214	1 |- ( -. A. x x = y -> ( z = y -> A. x z = y ) )

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:  ax12eq  34226  ax12el  34227  ax12inda  34233  ax12v2-o  34234