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Metamath Proof Explorer

Theorem speimfwALT

Description: Alternate proof of speimfw (longer compressed proof, but fewer essential steps). (Contributed by NM, 23-Apr-2017) (Proof shortened by Wolf Lammen, 5-Aug-2017) (Proof modification is discouraged.) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		speimfw.2	$⊢ x = y \to (φ \to ψ)$
2	1	eximi	$⊢ \exists x x = y \to \exists x (φ \to ψ)$
3		df-ex	$⊢ \exists x x = y \leftrightarrow \neg \forall x \neg x = y$
4		19.35	$⊢ \exists x (φ \to ψ) \leftrightarrow (\forall x φ \to \exists x ψ)$
5	2 3 4	3imtr3i	$⊢ \neg \forall x \neg x = y \to (\forall x φ \to \exists x ψ)$