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Mirrors > Home > MPE Home > Th. List > 19.28v | Structured version Visualization version GIF version |
Description: Version of 19.28 2221 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 1873 | . 2 ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓)) | |
2 | 19.3v 1985 | . . 3 ⊢ (∀𝑥𝜑 ↔ 𝜑) | |
3 | 2 | anbi1i 624 | . 2 ⊢ ((∀𝑥𝜑 ∧ ∀𝑥𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓)) |
4 | 1, 3 | bitri 274 | 1 ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 ∀wal 1537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1783 |
This theorem is referenced by: reu6 3661 dfer2 8499 kmlem14 9919 kmlem15 9920 bnj1176 32985 bnj1186 32987 ismnuprim 41912 19.28vv 42004 |
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