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Mirrors > Home > ILE Home > Th. List > albiim | 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 368 | . . 3 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑))) | |
2 | 1 | albii 1359 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) ↔ ∀𝑥((𝜑 → 𝜓) ∧ (𝜓 → 𝜑))) |
3 | 19.26 1370 | . 2 ⊢ (∀𝑥((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∀𝑥(𝜓 → 𝜑))) | |
4 | 2, 3 | bitri 173 | 1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∀𝑥(𝜓 → 𝜑))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∀wal 1241 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-gen 1338 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: 2albiim 1377 hbbid 1467 equveli 1642 spsbbi 1725 eu1 1925 eqss 2960 ssext 3957 |
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