Theorem ralsbii

Description: Congruence for "all some" restricted to a class. (Contributed by David A. Wheeler, 12-Jul-2026)

	Ref	Expression
Hypotheses	ralsbii.1	⊢ ( 𝜑 ↔ 𝜒 )
	ralsbii.2	⊢ ( 𝜓 ↔ 𝜃 )
Assertion	ralsbii	⊢ ( ∀∃ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ↔ ∀∃ 𝑥 ∈ 𝐴 ( 𝜒 → 𝜃 ) )

Proof

Step	Hyp	Ref	Expression
1		ralsbii.1	⊢ ( 𝜑 ↔ 𝜒 )
2		ralsbii.2	⊢ ( 𝜓 ↔ 𝜃 )
3	1 2	imbi12i	⊢ ( ( 𝜑 → 𝜓 ) ↔ ( 𝜒 → 𝜃 ) )
4	3	ralbii	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ↔ ∀ 𝑥 ∈ 𝐴 ( 𝜒 → 𝜃 ) )
5	1	rexbii	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ∃ 𝑥 ∈ 𝐴 𝜒 )
6	4 5	anbi12i	⊢ ( ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ∧ ∃ 𝑥 ∈ 𝐴 𝜑 ) ↔ ( ∀ 𝑥 ∈ 𝐴 ( 𝜒 → 𝜃 ) ∧ ∃ 𝑥 ∈ 𝐴 𝜒 ) )
7		df-rals	⊢ ( ∀∃ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ↔ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ∧ ∃ 𝑥 ∈ 𝐴 𝜑 ) )
8		df-rals	⊢ ( ∀∃ 𝑥 ∈ 𝐴 ( 𝜒 → 𝜃 ) ↔ ( ∀ 𝑥 ∈ 𝐴 ( 𝜒 → 𝜃 ) ∧ ∃ 𝑥 ∈ 𝐴 𝜒 ) )
9	6 7 8	3bitr4i	⊢ ( ∀∃ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ↔ ∀∃ 𝑥 ∈ 𝐴 ( 𝜒 → 𝜃 ) )