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Mirrors > Home > MPE Home > Th. List > nan | Structured version Visualization version GIF version |
Description: Theorem to move a conjunct in and out of a negation. (Contributed by NM, 9-Nov-2003.) |
Ref | Expression |
---|---|
nan | ⊢ ((𝜑 → ¬ (𝜓 ∧ 𝜒)) ↔ ((𝜑 ∧ 𝜓) → ¬ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | impexp 451 | . 2 ⊢ (((𝜑 ∧ 𝜓) → ¬ 𝜒) ↔ (𝜑 → (𝜓 → ¬ 𝜒))) | |
2 | imnan 400 | . . 3 ⊢ ((𝜓 → ¬ 𝜒) ↔ ¬ (𝜓 ∧ 𝜒)) | |
3 | 2 | imbi2i 335 | . 2 ⊢ ((𝜑 → (𝜓 → ¬ 𝜒)) ↔ (𝜑 → ¬ (𝜓 ∧ 𝜒))) |
4 | 1, 3 | bitr2i 275 | 1 ⊢ ((𝜑 → ¬ (𝜓 ∧ 𝜒)) ↔ ((𝜑 ∧ 𝜓) → ¬ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 206 df-an 397 |
This theorem is referenced by: pm4.15 831 somincom 6132 wemaplem2 9538 alephval3 10101 hauspwpwf1 23482 sticksstones22 40972 rtprmirr 41233 icccncfext 44589 stoweidlem34 44736 stirlinglem5 44780 fourierdlem42 44851 etransc 44985 |
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