Theorem ralsbii 50459

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

Hypotheses

Ref	Expression
ralsbii.1	⊢ (𝜑 ↔ 𝜒)
ralsbii.2	⊢ (𝜓 ↔ 𝜃)

Assertion

Ref	Expression
ralsbii	⊢ (∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ ∀∃𝑥 ∈ 𝐴(𝜒 → 𝜃))

Proof of Theorem ralsbii

Step	Hyp	Ref	Expression
1		ralsbii.1	. . . . 5 ⊢ (𝜑 ↔ 𝜒)
2		ralsbii.2	. . . . 5 ⊢ (𝜓 ↔ 𝜃)
3	1, 2	imbi12i 353	. . . 4 ⊢ ((𝜑 → 𝜓) ↔ (𝜒 → 𝜃))
4	3	ralbii 3117	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∀𝑥 ∈ 𝐴 (𝜒 → 𝜃))
5	1	rexbii 3118	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜒)
6	4, 5	anbi12i 639	. 2 ⊢ ((∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ∧ ∃𝑥 ∈ 𝐴 𝜑) ↔ (∀𝑥 ∈ 𝐴 (𝜒 → 𝜃) ∧ ∃𝑥 ∈ 𝐴 𝜒))
7		df-rals 50447	. 2 ⊢ (∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ∧ ∃𝑥 ∈ 𝐴 𝜑))
8		df-rals 50447	. 2 ⊢ (∀∃𝑥 ∈ 𝐴(𝜒 → 𝜃) ↔ (∀𝑥 ∈ 𝐴 (𝜒 → 𝜃) ∧ ∃𝑥 ∈ 𝐴 𝜒))
9	6, 7, 8	3bitr4i 306	1 ⊢ (∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ ∀∃𝑥 ∈ 𝐴(𝜒 → 𝜃))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wral 3085  ∃wrex 3095  ∀∃wrals 50445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-ral 3086  df-rex 3096  df-rals 50447

This theorem is referenced by: (None)