Theorem alsralrex 50618

Description: The general "all some" quantifier with class membership as its antecedent holds if and only if 𝜑 holds for every 𝑥 in 𝐴 and some 𝑥 in 𝐴 satisfies 𝜑. (Contributed by Peter Mazsa, 27-Nov-2018.) (Revised by David A. Wheeler, 15-Jul-2026.)

Assertion

Ref	Expression
alsralrex	⊢ (∀∃𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜑))

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem alsralrex

Step	Hyp	Ref	Expression
1		df-als 50594	. 2 ⊢ (∀∃𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ∧ ∃𝑥 𝑥 ∈ 𝐴))
2		df-ral 3079	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
3	2	bicomi 227	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑥 ∈ 𝐴 𝜑)
4	3	anbi1i 635	. . 3 ⊢ ((∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ∧ ∃𝑥 𝑥 ∈ 𝐴) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 𝑥 ∈ 𝐴))
5		n0 4306	. . . . . . . 8 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
6	5	biimpri 231	. . . . . . 7 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → 𝐴 ≠ ∅)
7		r19.2z 4459	. . . . . . 7 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝜑) → ∃𝑥 ∈ 𝐴 𝜑)
8	6, 7	sylan 591	. . . . . 6 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝜑) → ∃𝑥 ∈ 𝐴 𝜑)
9	8	expcom 418	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 𝜑))
10		rexn0 4456	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝐴 ≠ ∅)
11	5	biimpi 219	. . . . . . 7 ⊢ (𝐴 ≠ ∅ → ∃𝑥 𝑥 ∈ 𝐴)
12	10, 11	syl 18	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
13	12	a1i 11	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 𝑥 ∈ 𝐴))
14	9, 13	impbid 215	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝜑))
15	14	pm5.32i 584	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 𝑥 ∈ 𝐴) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜑))
16	4, 15	bitri 278	. 2 ⊢ ((∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ∧ ∃𝑥 𝑥 ∈ 𝐴) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜑))
17	1, 16	bitri 278	1 ⊢ (∀∃𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜑))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1567  ∃wex 1808   ∈ wcel 2142   ≠ wne 2957  ∀wral 3078  ∃wrex 3088  ∅c0 4285  ∀∃wals 50592

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-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-ne 2958  df-ral 3079  df-rex 3089  df-dif 3907  df-nul 4286  df-als 50594

This theorem is used by:  alsraln0  50619  ralals  50620  rexals  50621