Theorem alsraln0 50540

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 𝐴 is not empty. (Contributed by Peter Mazsa, 28-Nov-2018.) (Revised by David A. Wheeler, 15-Jul-2026.)

Assertion

Ref	Expression
alsraln0	⊢ (∀∃𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ 𝐴 ≠ ∅))

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem alsraln0

Step	Hyp	Ref	Expression
1		alsralrex 50539	. 2 ⊢ (∀∃𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜑))
2		rexn0 4462	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝐴 ≠ ∅)
3	2	a1i 11	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∃𝑥 ∈ 𝐴 𝜑 → 𝐴 ≠ ∅))
4		r19.2z 4465	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝜑) → ∃𝑥 ∈ 𝐴 𝜑)
5	4	expcom 418	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (𝐴 ≠ ∅ → ∃𝑥 ∈ 𝐴 𝜑))
6	3, 5	impbid 215	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∃𝑥 ∈ 𝐴 𝜑 ↔ 𝐴 ≠ ∅))
7	6	pm5.32i 584	. 2 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜑) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ 𝐴 ≠ ∅))
8	1, 7	bitri 278	1 ⊢ (∀∃𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ 𝐴 ≠ ∅))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∈ wcel 2150   ≠ wne 2965  ∀wral 3086  ∃wrex 3096  ∅c0 4294  ∀∃wals 50513

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-9 2160  ax-ext 2742

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-ne 2966  df-ral 3087  df-rex 3097  df-dif 3916  df-nul 4295  df-als 50515

This theorem is referenced by:  n0als  50543  2alsraln0  50544