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Metamath Proof Explorer

Theorem spimw

Description: Specialization. Lemma 8 of KalishMontague p. 87. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017) (Proof shortened by Wolf Lammen, 7-Aug-2017)

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

Proof

Step	Hyp	Ref	Expression
1		spimw.1	⊢ ( ¬ 𝜓 → ∀ 𝑥 ¬ 𝜓 )
2		spimw.2	⊢ ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) )
3		ax6v	⊢ ¬ ∀ 𝑥 ¬ 𝑥 = 𝑦
4	1 2	spimfw	⊢ ( ¬ ∀ 𝑥 ¬ 𝑥 = 𝑦 → ( ∀ 𝑥 𝜑 → 𝜓 ) )
5	3 4	ax-mp	⊢ ( ∀ 𝑥 𝜑 → 𝜓 )