Theorem dtrucor 5340

Description: Corollary of dtru 5416. 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 5341. (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 5339	. . 3 ⊢ ¬ ∀𝑥 𝑥 = 𝑦
2	1	pm2.21i 120	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑥 ≠ 𝑦)
3		dtrucor.1	. 2 ⊢ 𝑥 = 𝑦
4	2, 3	mpg 1830	1 ⊢ 𝑥 ≠ 𝑦

This proof depends on syntax axioms:  ∀wal 1568   ≠ wne 2957

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-nul 5267  ax-pow 5334

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813

This theorem is used by: (None)