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Mirrors > Home > MPE Home > Th. List > 19.28v | Structured version Visualization version GIF version |
Description: Version of 19.28 2213 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 1916 | . 2 ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓)) | |
2 | 19.3v 2031 | . . 3 ⊢ (∀𝑥𝜑 ↔ 𝜑) | |
3 | 2 | anbi1i 617 | . 2 ⊢ ((∀𝑥𝜑 ∧ ∀𝑥𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓)) |
4 | 1, 3 | bitri 267 | 1 ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 386 ∀wal 1599 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 |
This theorem depends on definitions: df-bi 199 df-an 387 df-ex 1824 |
This theorem is referenced by: reu6 3606 dfer2 8027 kmlem14 9320 kmlem15 9321 bnj1176 31686 bnj1186 31688 19.28vv 39533 |
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