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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 475 | . . 3 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑))) | |
2 | 1 | albii 1822 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) ↔ ∀𝑥((𝜑 → 𝜓) ∧ (𝜓 → 𝜑))) |
3 | 19.26 1873 | . 2 ⊢ (∀𝑥((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∀𝑥(𝜓 → 𝜑))) | |
4 | 2, 3 | bitri 274 | 1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∀𝑥(𝜓 → 𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∀wal 1537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 |
This theorem depends on definitions: df-bi 206 df-an 397 |
This theorem is referenced by: 2albiim 1893 dfmoeu 2536 mobi 2547 eu6lem 2573 eu1 2612 eqss 3936 ssext 5369 asymref2 6016 rabeqsnd 30834 pm14.122a 41999 |
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