Theorem dtrucor 5301

Description: Corollary of dtru 5377. 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 5302. (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		dtruALT2 5300	. . 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 1804	1 ⊢ 𝑥 ≠ 𝑦

Syntax hints:  ∀wal 1545   ≠ wne 2934

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-nul 5229  ax-pow 5295

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787

This theorem is referenced by: (None)