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Metamath Proof Explorer

Theorem speimfw

Description: Specialization, with additional weakening (compared to 19.2 ) 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, 5-Dec-2017)

		Ref	Expression
	Hypothesis	speimfw.2	$⊢ x = y \to (φ \to ψ)$
	Assertion	speimfw	$⊢ \neg \forall x \neg x = y \to (\forall x φ \to \exists x ψ)$

Proof

Step	Hyp	Ref	Expression
1		speimfw.2	$⊢ x = y \to (φ \to ψ)$
2		df-ex	$⊢ \exists x x = y \leftrightarrow \neg \forall x \neg x = y$
3	2	biimpri	$⊢ \neg \forall x \neg x = y \to \exists x x = y$
4	1	com12	$⊢ φ \to (x = y \to ψ)$
5	4	aleximi	$⊢ \forall x φ \to (\exists x x = y \to \exists x ψ)$
6	3 5	syl5com	$⊢ \neg \forall x \neg x = y \to (\forall x φ \to \exists x ψ)$