Theorem dtrucor 2741

Description: Corollary of dtru 2740. 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 2742.

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 2740	. . 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 962	1 |- x =/= y

Syntax hints:  A.wal 950   = wceq 1099   =/= wne 1561

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-13 1107  ax-14 1108  ax-11 1180  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436  ax-sep 2671  ax-nul 2678  ax-pow 2710

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 957  df-sb 1155  df-eu 1359  df-mo 1360  df-clab 1441  df-cleq 1446  df-clel 1449  df-ne 1563  df-v 1787  df-dif 2020  df-un 2021  df-in 2022  df-ss 2024  df-nul 2252  df-pw 2373  df-sn 2383  df-pr 2384

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