Metamath Proof Explorer


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 )