Theorem rexals

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

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

Proof

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