Theorem dveel2ALT 34224

Description: Alternate proof of dveel2 2371 using ax-c16 34177 instead of ax-5 1839. (Contributed by NM, 10-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
dveel2ALT	|- ( -. A. x x = y -> ( z e. y -> A. x z e. y ) )

Distinct variable group:   x, z

Proof of Theorem dveel2ALT

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax5el 34222	. 2 |- ( z e. w -> A. x z e. w )
2		ax5el 34222	. 2 |- ( z e. y -> A. w z e. y )
3		elequ2 2004	. 2 |- ( w = y -> ( z e. w <-> z e. y ) )
4	1, 2, 3	dvelimh 2336	1 |- ( -. A. x x = y -> ( z e. y -> A. x z e. 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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-c14 34176  ax-c16 34177

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: (None)