Mathbox for Rohan Ridenour |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > expandan | Structured version Visualization version GIF version |
Description: Expand conjunction to primitives. (Contributed by Rohan Ridenour, 13-Aug-2023.) |
Ref | Expression |
---|---|
expandan.1 | ⊢ (𝜑 ↔ 𝜓) |
expandan.2 | ⊢ (𝜒 ↔ 𝜃) |
Ref | Expression |
---|---|
expandan | ⊢ ((𝜑 ∧ 𝜒) ↔ ¬ (𝜓 → ¬ 𝜃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | expandan.1 | . . 3 ⊢ (𝜑 ↔ 𝜓) | |
2 | expandan.2 | . . 3 ⊢ (𝜒 ↔ 𝜃) | |
3 | 1, 2 | anbi12i 630 | . 2 ⊢ ((𝜑 ∧ 𝜒) ↔ (𝜓 ∧ 𝜃)) |
4 | df-an 400 | . 2 ⊢ ((𝜓 ∧ 𝜃) ↔ ¬ (𝜓 → ¬ 𝜃)) | |
5 | 3, 4 | bitri 278 | 1 ⊢ ((𝜑 ∧ 𝜒) ↔ ¬ (𝜓 → ¬ 𝜃)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 |
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 210 df-an 400 |
This theorem is referenced by: ismnuprim 41613 rr-grothprimbi 41614 |
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