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Metamath Proof Explorer

Theorem spimfv

Description: Specialization, using implicit substitution. Version of spim with a disjoint variable condition, which does not require ax-13 . See spimvw for a version with two disjoint variable conditions, requiring fewer axioms, and spimv for another variant. (Contributed by NM, 10-Jan-1993) (Revised by BJ, 31-May-2019)

	Ref	Expression
Hypotheses	spimfv.nf	$⊢ Ⅎ x ψ$
	spimfv.1	$⊢ x = y \to (φ \to ψ)$
Assertion	spimfv	$⊢ \forall x φ \to ψ$

Proof

Step	Hyp	Ref	Expression
1		spimfv.nf	$⊢ Ⅎ x ψ$
2		spimfv.1	$⊢ x = y \to (φ \to ψ)$
3		ax6ev	$⊢ \exists x x = y$
4	3 2	eximii	$⊢ \exists x (φ \to ψ)$
5	1 4	19.36i	$⊢ \forall x φ \to ψ$