Theorem nd4 9412

Description: A lemma for proving conditionless ZFC axioms. (Contributed by NM, 2-Jan-2002.)

Assertion

Ref	Expression
nd4	|- ( A. x x = y -> -. A. z y e. x )

Proof of Theorem nd4

Step	Hyp	Ref	Expression
1		nd3 9411	. 2 |- ( A. y y = x -> -. A. z y e. x )
2	1	aecoms 2312	1 |- ( A. x x = y -> -. A. z y e. x )

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-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-reg 8497

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-un 3579  df-nul 3916  df-sn 4178  df-pr 4180

This theorem is referenced by:  axrepnd  9416