Theorem equsexv 2269

Description: An equivalence related to implicit substitution. Version of equsex 2419 with a disjoint variable condition, which does not require ax-13 2373. See equsexvw 2016 for a version with two disjoint variable conditions requiring fewer axioms. See also the dual form equsalv 2268. (Contributed by NM, 5-Aug-1993.) (Revised by BJ, 31-May-2019.)

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		sbalex 2244	. 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
2		equsalv.nf	. . 3 ⊢ Ⅎ𝑥𝜓
3		equsalv.1	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	2, 3	equsalv 2268	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓)
5	1, 4	bitri 278	1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1540  ∃wex 1786  Ⅎwnf 1790

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-10 2145  ax-12 2179

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-ex 1787  df-nf 1791

This theorem is referenced by:  sb5OLD  2277  equsexhv  2297  cleljustALT2  2366  sb10f  2533  dprd2d2  19297  poimirlem25  35457