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 = 𝑦 ∧ 𝜑 ) → ∀ 𝑥 ( 𝑥 = 𝑦 → ∃ 𝑦 𝜑 ) )

Proof

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