Theorem hbsb2e

Description: Special case of a bound-variable hypothesis builder for substitution. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 2-Feb-2007) (New usage is discouraged.)

	Ref	Expression
Assertion	hbsb2e	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 → ∀ 𝑥 [ 𝑦 / 𝑥 ] ∃ 𝑦 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		sb4e	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → ∃ 𝑦 𝜑 ) )
2		sb2	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → ∃ 𝑦 𝜑 ) → [ 𝑦 / 𝑥 ] ∃ 𝑦 𝜑 )
3	2	axc4i	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → ∃ 𝑦 𝜑 ) → ∀ 𝑥 [ 𝑦 / 𝑥 ] ∃ 𝑦 𝜑 )
4	1 3	syl	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 → ∀ 𝑥 [ 𝑦 / 𝑥 ] ∃ 𝑦 𝜑 )