|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > ralbiim | Structured version Visualization version GIF version | ||
| Description: Split a biconditional and distribute quantifier. Restricted quantifier version of albiim 1888. (Contributed by NM, 3-Jun-2012.) | 
| Ref | Expression | 
|---|---|
| ralbiim | ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) ↔ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | dfbi2 474 | . . 3 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑))) | |
| 2 | 1 | ralbii 3092 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) ↔ ∀𝑥 ∈ 𝐴 ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑))) | 
| 3 | r19.26 3110 | . 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))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wral 3060 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ral 3061 | 
| This theorem is referenced by: 2ralbiim 3131 eqreu 3734 isclo2 23097 chrelat4i 32393 hlateq 39402 ntrneik13 44116 ntrneix13 44117 | 
| Copyright terms: Public domain | W3C validator |