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Metamath Proof Explorer

Theorem spimvALT

Description: Alternate proof of spimv . Note that it requires only ax-1 through ax-5 together with ax6e . Currently, proofs derive from ax6v , but if ax-6 could be used instead, this proof would reduce axiom usage. (Contributed by NM, 31-Jul-1993) Remove dependency on ax-10 . (Revised by BJ, 29-Nov-2020) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Hypothesis	spimv.1	$⊢ x = y \to (φ \to ψ)$
	Assertion	spimvALT	$⊢ \forall x φ \to ψ$

Proof

Step	Hyp	Ref	Expression
1		spimv.1	$⊢ x = y \to (φ \to ψ)$
2		ax6e	$⊢ \exists x x = y$
3	2 1	eximii	$⊢ \exists x (φ \to ψ)$
4	3	19.36iv	$⊢ \forall x φ \to ψ$