Theorem equs5a

Description: A property related to substitution that unlike equs5 does not require a distinctor antecedent. Usage of this theorem is discouraged because it depends on ax-13 . This proof uses ax12 , see equs5aALT for an alternative one using ax-12 but not ax13 . Usage of the weaker equs5av is preferred, which uses ax12v2 , but not ax-13 . (Contributed by NM, 2-Feb-2007) (New usage is discouraged.)

	Ref	Expression
Assertion	equs5a	⊢ ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ ∀ 𝑦 𝜑 ) → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		nfa1	⊢ Ⅎ 𝑥 ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 )
2		ax12	⊢ ( 𝑥 = 𝑦 → ( ∀ 𝑦 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) )
3	2	imp	⊢ ( ( 𝑥 = 𝑦 ∧ ∀ 𝑦 𝜑 ) → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) )
4	1 3	exlimi	⊢ ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ ∀ 𝑦 𝜑 ) → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) )