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Metamath Proof Explorer

Theorem bj-hbs1

Description: Version of hbsb2 with a disjoint variable condition, which does not require ax-13 , and removal of ax-13 from hbs1 . (Contributed by BJ, 23-Jun-2019) (Proof modification is discouraged.)

		Ref	Expression
	Assertion	bj-hbs1	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 → ∀ 𝑥 [ 𝑦 / 𝑥 ] 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		sb6	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) )
2	1	biimpri	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) → [ 𝑦 / 𝑥 ] 𝜑 )
3	2	axc4i	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) → ∀ 𝑥 [ 𝑦 / 𝑥 ] 𝜑 )
4	1 3	sylbi	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 → ∀ 𝑥 [ 𝑦 / 𝑥 ] 𝜑 )