Theorem equsexh 2452

Description: An equivalence related to implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2403. See equsexhv 2326 for a version with a disjoint variable condition which does not require ax-13 2403. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Hypotheses

Ref	Expression
equsalh.1	⊢ (𝜓 → ∀𝑥𝜓)
equsalh.2	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
equsexh	⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓)

Proof of Theorem equsexh

Step	Hyp	Ref	Expression
1		equsalh.1	. . 3 ⊢ (𝜓 → ∀𝑥𝜓)
2	1	nf5i 2180	. 2 ⊢ Ⅎ𝑥𝜓
3		equsalh.2	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	2, 3	equsex 2449	1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1567  ∃wex 1808

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-10 2175  ax-12 2212  ax-13 2403

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1809  df-nf 1813

This theorem is used by: (None)