Theorem dvdemo2 2771

Description: Demonstration of a theorem that requires x and z to be distinct, but no others. Compare dvdemo1 2770.

Assertion

Ref	Expression
dvdemo2	|- E.x(x = y -> z e. x)

Distinct variable group:   x,z

Proof of Theorem dvdemo2

Step	Hyp	Ref	Expression
1		el 2746	. 2 |- E.x z e. x
2		ax-1 4	. . 3 |- (z e. x -> (x = y -> z e. x))
3	2	19.22i 1038	. 2 |- (E.x z e. x -> E.x(x = y -> z e. x))
4	1, 3	ax-mp 7	1 |- E.x(x = y -> z e. x)

Syntax hints:   -> wi 3   = wceq 954   e. wcel 956  E.wex 978

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409

Copyright terms: Public domain