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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 → ( 𝜑 → 𝜓 ) )
Assertion	spimfv	⊢ ( ∀ 𝑥 𝜑 → 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		spimfv.nf	⊢ Ⅎ 𝑥 𝜓
2		spimfv.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) )
3		ax6ev	⊢ ∃ 𝑥 𝑥 = 𝑦
4	3 2	eximii	⊢ ∃ 𝑥 ( 𝜑 → 𝜓 )
5	1 4	19.36i	⊢ ( ∀ 𝑥 𝜑 → 𝜓 )