Theorem hbsb3

Description: If y is not free in ph , x is not free in [ y / x ] ph . Usage of this theorem is discouraged because it depends on ax-13 . Check out bj-hbsb3v for a weaker version requiring less axioms. (Contributed by NM, 14-May-1993) (New usage is discouraged.)

	Ref	Expression
Hypothesis	hbsb3.1	⊢ ( 𝜑 → ∀ 𝑦 𝜑 )
Assertion	hbsb3	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 → ∀ 𝑥 [ 𝑦 / 𝑥 ] 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		hbsb3.1	⊢ ( 𝜑 → ∀ 𝑦 𝜑 )
2	1	sbimi	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 → [ 𝑦 / 𝑥 ] ∀ 𝑦 𝜑 )
3		hbsb2a	⊢ ( [ 𝑦 / 𝑥 ] ∀ 𝑦 𝜑 → ∀ 𝑥 [ 𝑦 / 𝑥 ] 𝜑 )
4	2 3	syl	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 → ∀ 𝑥 [ 𝑦 / 𝑥 ] 𝜑 )