Theorem dtrucor 2769

Description: Corollary of dtru 2768. 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 2770.

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 2768	. . 3 ⊢ ¬ ∀x x = y
2	1	pm2.21i 77	. 2 ⊢ (∀x x = y → x ≠ y)
3		dtrucor.1	. 2 ⊢ x = y
4	2, 3	mpg 984	1 ⊢ x ≠ y

Syntax hints:  ∀wal 952   = wceq 954   ≠ wne 1582

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 2699  ax-nul 2706  ax-pow 2738

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

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