Theorem equsexh

Description: An equivalence related to implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . See equsexhv for a version with a disjoint variable condition which does not require ax-13 . (Contributed by NM, 5-Aug-1993) (New usage is discouraged.)

	Ref	Expression
Hypotheses	equsalh.1	⊢ ( 𝜓 → ∀ 𝑥 𝜓 )
	equsalh.2	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
Assertion	equsexh	⊢ ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ↔ 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		equsalh.1	⊢ ( 𝜓 → ∀ 𝑥 𝜓 )
2		equsalh.2	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
3	1	nf5i	⊢ Ⅎ 𝑥 𝜓
4	3 2	equsex	⊢ ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ↔ 𝜓 )