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Mirrors > Home > MPE Home > Th. List > Mathboxes > simplbi2comtVD | Structured version Visualization version GIF version |
Description: Virtual deduction proof of simplbi2comt 501.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
simplbi2comt 501 is simplbi2comtVD 44886 without virtual deductions and was
automatically derived from simplbi2comtVD 44886.
|
Ref | Expression |
---|---|
simplbi2comtVD | ⊢ ((𝜑 ↔ (𝜓 ∧ 𝜒)) → (𝜒 → (𝜓 → 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idn1 44572 | . . . . 5 ⊢ ( (𝜑 ↔ (𝜓 ∧ 𝜒)) ▶ (𝜑 ↔ (𝜓 ∧ 𝜒)) ) | |
2 | biimpr 220 | . . . . 5 ⊢ ((𝜑 ↔ (𝜓 ∧ 𝜒)) → ((𝜓 ∧ 𝜒) → 𝜑)) | |
3 | 1, 2 | e1a 44625 | . . . 4 ⊢ ( (𝜑 ↔ (𝜓 ∧ 𝜒)) ▶ ((𝜓 ∧ 𝜒) → 𝜑) ) |
4 | pm3.3 448 | . . . 4 ⊢ (((𝜓 ∧ 𝜒) → 𝜑) → (𝜓 → (𝜒 → 𝜑))) | |
5 | 3, 4 | e1a 44625 | . . 3 ⊢ ( (𝜑 ↔ (𝜓 ∧ 𝜒)) ▶ (𝜓 → (𝜒 → 𝜑)) ) |
6 | pm2.04 90 | . . 3 ⊢ ((𝜓 → (𝜒 → 𝜑)) → (𝜒 → (𝜓 → 𝜑))) | |
7 | 5, 6 | e1a 44625 | . 2 ⊢ ( (𝜑 ↔ (𝜓 ∧ 𝜒)) ▶ (𝜒 → (𝜓 → 𝜑)) ) |
8 | 7 | in1 44569 | 1 ⊢ ((𝜑 ↔ (𝜓 ∧ 𝜒)) → (𝜒 → (𝜓 → 𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 |
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 207 df-an 396 df-vd1 44568 |
This theorem is referenced by: (None) |
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