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Metamath Proof Explorer

Theorem spimfw

Description: Specialization, with additional weakening (compared to sp ) to allow bundling of x and y . Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-2017) (Proof shortened by Wolf Lammen, 7-Aug-2017)

	Ref	Expression
Hypotheses	spimfw.1	⊢ ( ¬ 𝜓 → ∀ 𝑥 ¬ 𝜓 )
	spimfw.2	⊢ ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) )
Assertion	spimfw	⊢ ( ¬ ∀ 𝑥 ¬ 𝑥 = 𝑦 → ( ∀ 𝑥 𝜑 → 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		spimfw.1	⊢ ( ¬ 𝜓 → ∀ 𝑥 ¬ 𝜓 )
2		spimfw.2	⊢ ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) )
3	2	speimfw	⊢ ( ¬ ∀ 𝑥 ¬ 𝑥 = 𝑦 → ( ∀ 𝑥 𝜑 → ∃ 𝑥 𝜓 ) )
4		df-ex	⊢ ( ∃ 𝑥 𝜓 ↔ ¬ ∀ 𝑥 ¬ 𝜓 )
5	1	con1i	⊢ ( ¬ ∀ 𝑥 ¬ 𝜓 → 𝜓 )
6	4 5	sylbi	⊢ ( ∃ 𝑥 𝜓 → 𝜓 )
7	3 6	syl6	⊢ ( ¬ ∀ 𝑥 ¬ 𝑥 = 𝑦 → ( ∀ 𝑥 𝜑 → 𝜓 ) )