Theorem dtrucor 2763

Description: Corollary of dtru 2762. This example illustrates the danger of blindly trusting the standard Deduction Theorem without accounting for free variables: the theorem form of this deduction is not valid, as shown by dtrucor2 2764.

Hypothesis

Ref	Expression
dtrucor.1	|- x = y

Assertion

Ref	Expression
dtrucor	|- x =/= y

Distinct variable group:   x,y

Proof of Theorem dtrucor

Step	Hyp	Ref	Expression
1		dtru 2762	. . 3 |- -. A.x x = y
2	1	pm2.21i 77	. 2 |- (A.x x = y -> x =/= y)
3		dtrucor.1	. 2 |- x = y
4	2, 3	mpg 983	1 |- x =/= y

Syntax hints:  A.wal 951   = wceq 953   =/= wne 1577

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-nul 2700  ax-pow 2732

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403

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