Metamath Proof Explorer


Theorem hbsb2

Description: Bound-variable hypothesis builder for substitution. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 14-May-1993) (New usage is discouraged.)

Ref Expression
Assertion hbsb2 ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( [ 𝑦 / 𝑥 ] 𝜑 → ∀ 𝑥 [ 𝑦 / 𝑥 ] 𝜑 ) )

Proof

Step Hyp Ref Expression
1 sb4b ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( [ 𝑦 / 𝑥 ] 𝜑 ↔ ∀ 𝑥 ( 𝑥 = 𝑦𝜑 ) ) )
2 sb2 ( ∀ 𝑥 ( 𝑥 = 𝑦𝜑 ) → [ 𝑦 / 𝑥 ] 𝜑 )
3 2 axc4i ( ∀ 𝑥 ( 𝑥 = 𝑦𝜑 ) → ∀ 𝑥 [ 𝑦 / 𝑥 ] 𝜑 )
4 1 3 syl6bi ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( [ 𝑦 / 𝑥 ] 𝜑 → ∀ 𝑥 [ 𝑦 / 𝑥 ] 𝜑 ) )