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| Mirrors > Home > MPE Home > Th. List > albiim | Structured version Visualization version GIF version | ||
| Description: Split a biconditional and distribute quantifier. (Contributed by NM, 18-Aug-1993.) |
| Ref | Expression |
|---|---|
| albiim | ⊢ (∀𝑥(𝜑 ↔ 𝜓) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∀𝑥(𝜓 → 𝜑))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dfbi2 474 | . . 3 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑))) | |
| 2 | 1 | albii 1819 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) ↔ ∀𝑥((𝜑 → 𝜓) ∧ (𝜓 → 𝜑))) |
| 3 | 19.26 1870 | . 2 ⊢ (∀𝑥((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∀𝑥(𝜓 → 𝜑))) | |
| 4 | 2, 3 | bitri 275 | 1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∀𝑥(𝜓 → 𝜑))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1538 |
| 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 |
| This theorem is referenced by: 2albiim 1890 dfmoeu 2536 mobi 2547 eu6lem 2573 eu1 2610 eqss 3999 rabeqsnd 4669 ssext 5459 asymref2 6137 eu6w 42686 pm14.122a 44441 |
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