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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	$⊢ \neg ψ \to \forall x \neg ψ$
	spimw.2	$⊢ x = y \to (φ \to ψ)$
Assertion	spimw	$⊢ \forall x φ \to ψ$

Proof

Step	Hyp	Ref	Expression
1		spimw.1	$⊢ \neg ψ \to \forall x \neg ψ$
2		spimw.2	$⊢ x = y \to (φ \to ψ)$
3		ax6v	$⊢ \neg \forall x \neg x = y$
4	1 2	spimfw	$⊢ \neg \forall x \neg x = y \to (\forall x φ \to ψ)$
5	3 4	ax-mp	$⊢ \forall x φ \to ψ$