Theorem equsexh 2295

Description: An equivalence related to implicit substitution. See equsexhv 2124 for a version with a dv condition which does not require ax-13 2246. (Contributed by NM, 5-Aug-1993.)

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 2024	. 2 ⊢ Ⅎ𝑥𝜓
3		equsalh.2	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	2, 3	equsex 2292	1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1481  ∃wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710

This theorem is referenced by: (None)