Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > 19.26-2 | Structured version Visualization version GIF version |
Description: Theorem 19.26 1862 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 1862 | . . 3 ⊢ (∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑦𝜑 ∧ ∀𝑦𝜓)) | |
2 | 1 | albii 1811 | . 2 ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ ∀𝑥(∀𝑦𝜑 ∧ ∀𝑦𝜓)) |
3 | 19.26 1862 | . 2 ⊢ (∀𝑥(∀𝑦𝜑 ∧ ∀𝑦𝜓) ↔ (∀𝑥∀𝑦𝜑 ∧ ∀𝑥∀𝑦𝜓)) | |
4 | 2, 3 | bitri 276 | 1 ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑥∀𝑦𝜑 ∧ ∀𝑥∀𝑦𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 ∀wal 1526 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 |
This theorem depends on definitions: df-bi 208 df-an 397 |
This theorem is referenced by: 2mo2 2725 opelopabt 5410 fun11 6421 dford4 39504 undmrnresiss 39842 ichan 43507 |
Copyright terms: Public domain | W3C validator |