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Metamath Proof Explorer

Theorem dtrucor2

Description: The theorem form of the deduction dtrucor leads to a contradiction, as mentioned in the "Wrong!" example at mmdeduction.html#bad . Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 20-Oct-2007) (New usage is discouraged.)

		Ref	Expression
	Hypothesis	dtrucor2.1	⊢ ( 𝑥 = 𝑦 → 𝑥 ≠ 𝑦 )
	Assertion	dtrucor2	⊢ ( 𝜑 ∧ ¬ 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		dtrucor2.1	⊢ ( 𝑥 = 𝑦 → 𝑥 ≠ 𝑦 )
2		ax6e	⊢ ∃ 𝑥 𝑥 = 𝑦
3	1	necon2bi	⊢ ( 𝑥 = 𝑦 → ¬ 𝑥 = 𝑦 )
4		pm2.01	⊢ ( ( 𝑥 = 𝑦 → ¬ 𝑥 = 𝑦 ) → ¬ 𝑥 = 𝑦 )
5	3 4	ax-mp	⊢ ¬ 𝑥 = 𝑦
6	5	nex	⊢ ¬ ∃ 𝑥 𝑥 = 𝑦
7	2 6	pm2.24ii	⊢ ( 𝜑 ∧ ¬ 𝜑 )