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Metamath Proof Explorer

Theorem bj-hbsb3v

Description: Version of hbsb3 with a disjoint variable condition, which does not require ax-13 . (Remark: the unbundled version of nfs1 is given by bj-nfs1v .) (Contributed by BJ, 11-Sep-2019) (Proof modification is discouraged.)

		Ref	Expression
	Hypothesis	bj-hbsb3v.1	⊢ ( 𝜑 → ∀ 𝑦 𝜑 )
	Assertion	bj-hbsb3v	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 → ∀ 𝑥 [ 𝑦 / 𝑥 ] 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		bj-hbsb3v.1	⊢ ( 𝜑 → ∀ 𝑦 𝜑 )
2	1	sbimi	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 → [ 𝑦 / 𝑥 ] ∀ 𝑦 𝜑 )
3		bj-hbsb2av	⊢ ( [ 𝑦 / 𝑥 ] ∀ 𝑦 𝜑 → ∀ 𝑥 [ 𝑦 / 𝑥 ] 𝜑 )
4	2 3	syl	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 → ∀ 𝑥 [ 𝑦 / 𝑥 ] 𝜑 )