Theorem equsexv 2254

Description: An equivalence related to implicit substitution. Version of equsex 2411 with a disjoint variable condition, which does not require ax-13 2365. See equsexvw 2000 for a version with two disjoint variable conditions requiring fewer axioms. See also the dual form equsalv 2253. (Contributed by NM, 5-Aug-1993.) (Revised by BJ, 31-May-2019.) Avoid ax-10 2129. (Revised by Gino Giotto, 18-Nov-2024.)

Hypotheses

Ref	Expression
equsalv.nf	⊢ Ⅎ𝑥𝜓
equsalv.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

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

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem equsexv

Step	Hyp	Ref	Expression
1		equsalv.nf	. . 3 ⊢ Ⅎ𝑥𝜓
2		equsalv.1	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3	2	biimpa 475	. . 3 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → 𝜓)
4	1, 3	exlimi 2205	. 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → 𝜓)
5	1, 2	equsalv 2253	. . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓)
6		equs4v 1995	. . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
7	5, 6	sylbir 234	. 2 ⊢ (𝜓 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
8	4, 7	impbii 208	1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394  ∀wal 1531  ∃wex 1773  Ⅎwnf 1777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-12 2166

This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1774  df-nf 1778

This theorem is referenced by:  sb5OLD  2263  equsexhv  2281  cleljustALT2  2356  sb10f  2520  dprd2d2  20013  poimirlem25  37249