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| Mirrors > Home > MPE Home > Th. List > 19.26-2 | Structured version Visualization version GIF version | ||
| Description: Theorem 19.26 1869 with two quantifiers. (Contributed by NM, 3-Feb-2005.) | 
| Ref | Expression | 
|---|---|
| 19.26-2 | ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑥∀𝑦𝜑 ∧ ∀𝑥∀𝑦𝜓)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | 19.26 1869 | . . 3 ⊢ (∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑦𝜑 ∧ ∀𝑦𝜓)) | |
| 2 | 1 | albii 1818 | . 2 ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ ∀𝑥(∀𝑦𝜑 ∧ ∀𝑦𝜓)) | 
| 3 | 19.26 1869 | . 2 ⊢ (∀𝑥(∀𝑦𝜑 ∧ ∀𝑦𝜓) ↔ (∀𝑥∀𝑦𝜑 ∧ ∀𝑥∀𝑦𝜓)) | |
| 4 | 2, 3 | bitri 275 | 1 ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑥∀𝑦𝜑 ∧ ∀𝑥∀𝑦𝜓)) | 
| 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 | 
| This theorem depends on definitions: df-bi 207 df-an 396 | 
| This theorem is referenced by: aaan 2332 2mo2 2646 opelopabt 5536 fun11 6639 dford4 43046 undmrnresiss 43622 | 
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