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| Mirrors > Home > MPE Home > Th. List > 19.28v | Structured version Visualization version GIF version | ||
| Description: Version of 19.28 2227 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 25-Mar-2004.) | 
| Ref | Expression | 
|---|---|
| 19.28v | ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | 19.26 1869 | . 2 ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓)) | |
| 2 | 19.3v 1980 | . 2 ⊢ (∀𝑥𝜑 ↔ 𝜑) | |
| 3 | 1, 2 | bianbi 627 | 1 ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 ∀wal 1537 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1779 | 
| This theorem is referenced by: reu6 3731 dfer2 8747 kmlem14 10205 kmlem15 10206 bnj1176 35020 bnj1186 35022 ismnuprim 44318 19.28vv 44410 | 
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