Theorem dtrucor 5264

Description: Corollary of dtru 5263. 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 5265. (Contributed by NM, 27-Jun-2002.)

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 5263	. . 3 ⊢ ¬ ∀𝑥 𝑥 = 𝑦
2	1	pm2.21i 119	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑥 ≠ 𝑦)
3		dtrucor.1	. 2 ⊢ 𝑥 = 𝑦
4	2, 3	mpg 1794	1 ⊢ 𝑥 ≠ 𝑦

Syntax hints:  ∀wal 1531   ≠ wne 3016

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-12 2173  ax-nul 5202  ax-pow 5258

This theorem depends on definitions:  df-bi 209  df-an 399  df-tru 1536  df-ex 1777  df-nf 1781

This theorem is referenced by: (None)