Theorem rexals 50542

Description: If some 𝑥 in 𝐴 satisfies 𝜑, then the general "all some" quantifier with class membership as its antecedent reduces to the assertion that 𝜑 holds for every 𝑥 in 𝐴. See rexrals 50536 for the restricted counterpart. (Contributed by Peter Mazsa, 19-Dec-2018.) (Revised by David A. Wheeler, 15-Jul-2026.)

Assertion

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

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rexals

Step	Hyp	Ref	Expression
1		alsralrex 50539	. 2 ⊢ (∀∃𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜑))
2		iba 536	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → (∀𝑥 ∈ 𝐴 𝜑 ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜑)))
3	2	bicomd 226	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜑) ↔ ∀𝑥 ∈ 𝐴 𝜑))
4	1, 3	bitrid 286	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → (∀∃𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑥 ∈ 𝐴 𝜑))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∈ wcel 2150  ∀wral 3086  ∃wrex 3096  ∀∃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: (None)