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

Proof

Step	Hyp	Ref	Expression
1		spimfw.1	$⊢ \neg ψ \to \forall x \neg ψ$
2		spimfw.2	$⊢ x = y \to (φ \to ψ)$
3	2	speimfw	$⊢ \neg \forall x \neg x = y \to (\forall x φ \to \exists x ψ)$
4		df-ex	$⊢ \exists x ψ \leftrightarrow \neg \forall x \neg ψ$
5	1	con1i	$⊢ \neg \forall x \neg ψ \to ψ$
6	4 5	sylbi	$⊢ \exists x ψ \to ψ$
7	3 6	syl6	$⊢ \neg \forall x \neg x = y \to (\forall x φ \to ψ)$