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Metamath Proof Explorer

Theorem spimvw

Description: A weak form of specialization. Lemma 8 of KalishMontague p. 87. Uses only Tarski's FOL axiom schemes. For stronger forms using more axioms, see spimv and spimfv . (Contributed by NM, 9-Apr-2017)

		Ref	Expression
	Hypothesis	spimvw.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) )
	Assertion	spimvw	⊢ ( ∀ 𝑥 𝜑 → 𝜓 )

Proof

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