Theorem ralbiim 3101

Description: Split a biconditional and distribute quantifier. Restricted quantifier version of albiim 1889. (Contributed by NM, 3-Jun-2012.)

Assertion

Ref	Expression
ralbiim	⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) ↔ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑)))

Proof of Theorem ralbiim

Step	Hyp	Ref	Expression
1		dfbi2 474	. . 3 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)))
2	1	ralbii 3083	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) ↔ ∀𝑥 ∈ 𝐴 ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)))
3		r19.26 3099	. 2 ⊢ (∀𝑥 ∈ 𝐴 ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)) ↔ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑)))
4	2, 3	bitri 275	1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) ↔ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wral 3052

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809

This theorem depends on definitions:  df-bi 207  df-an 396  df-ral 3053

This theorem is referenced by:  2ralbiim  3120  eqreu  3717  isclo2  23031  chrelat4i  32359  hlateq  39423  ntrneik13  44089  ntrneix13  44090