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Mirrors > Home > MPE Home > Th. List > 19.28v | Structured version Visualization version GIF version |
Description: Version of 19.28 2220 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 1862 | . 2 ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓)) | |
2 | 19.3v 1977 | . . 3 ⊢ (∀𝑥𝜑 ↔ 𝜑) | |
3 | 2 | anbi1i 623 | . 2 ⊢ ((∀𝑥𝜑 ∧ ∀𝑥𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓)) |
4 | 1, 3 | bitri 276 | 1 ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓)) |
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 ax-5 1902 ax-6 1961 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 |
This theorem is referenced by: reu6 3714 dfer2 8279 kmlem14 9577 kmlem15 9578 bnj1176 32174 bnj1186 32176 ismnuprim 40507 19.28vv 40595 |
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