Metamath Proof Explorer


Theorem equs5e

Description: A property related to substitution that unlike equs5 does not require a distinctor antecedent. This proof uses ax12 , see equs5eALT for an alternative one using ax-12 but not ax13 . Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 2-Feb-2007) (Proof shortened by Wolf Lammen, 15-Jan-2018) (New usage is discouraged.)

Ref Expression
Assertion equs5e ( ∃ 𝑥 ( 𝑥 = 𝑦𝜑 ) → ∀ 𝑥 ( 𝑥 = 𝑦 → ∃ 𝑦 𝜑 ) )

Proof

Step Hyp Ref Expression
1 nfa1 𝑥𝑥 ( 𝑥 = 𝑦 → ∃ 𝑦 𝜑 )
2 ax12 ( 𝑥 = 𝑦 → ( ∀ 𝑦𝑦 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → ∃ 𝑦 𝜑 ) ) )
3 hbe1 ( ∃ 𝑦 𝜑 → ∀ 𝑦𝑦 𝜑 )
4 3 19.23bi ( 𝜑 → ∀ 𝑦𝑦 𝜑 )
5 2 4 impel ( ( 𝑥 = 𝑦𝜑 ) → ∀ 𝑥 ( 𝑥 = 𝑦 → ∃ 𝑦 𝜑 ) )
6 1 5 exlimi ( ∃ 𝑥 ( 𝑥 = 𝑦𝜑 ) → ∀ 𝑥 ( 𝑥 = 𝑦 → ∃ 𝑦 𝜑 ) )