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Mirrors > Home > MPE Home > Th. List > 19.28v | Structured version Visualization version GIF version |
Description: Version of 19.28 2083 with a dv 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 1786 | . 2 ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓)) | |
2 | 19.3v 1884 | . . 3 ⊢ (∀𝑥𝜑 ↔ 𝜑) | |
3 | 2 | anbi1i 727 | . 2 ⊢ ((∀𝑥𝜑 ∧ ∀𝑥𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓)) |
4 | 1, 3 | bitri 263 | 1 ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 ∀wal 1473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 |
This theorem is referenced by: reu6 3362 dfer2 7630 kmlem14 8868 kmlem15 8869 bnj1176 30327 bnj1186 30329 19.28vv 37607 |
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