Theorem ralbiim 3172

Description: Split a biconditional and distribute quantifier. Restricted quantifier version of albiim 1884. (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 477	. . 3 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)))
2	1	ralbii 3163	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) ↔ ∀𝑥 ∈ 𝐴 ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)))
3		r19.26 3168	. 2 ⊢ (∀𝑥 ∈ 𝐴 ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)) ↔ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑)))
4	2, 3	bitri 277	1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) ↔ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wral 3136

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804

This theorem depends on definitions:  df-bi 209  df-an 399  df-ral 3141

This theorem is referenced by:  eqreu  3718  isclo2  21688  chrelat4i  30142  hlateq  36527  ntrneik13  40438  ntrneix13  40439  2ralbiim  43293