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Mirrors > Home > MPE Home > Th. List > Mathboxes > simplbi2comtVD | Structured version Visualization version GIF version |
Description: Virtual deduction proof of simplbi2comt 654.
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 654 is simplbi2comtVD 38146 without virtual deductions and was
automatically derived from simplbi2comtVD 38146.
|
Ref | Expression |
---|---|
simplbi2comtVD | ⊢ ((𝜑 ↔ (𝜓 ∧ 𝜒)) → (𝜒 → (𝜓 → 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idn1 37811 | . . . . 5 ⊢ ( (𝜑 ↔ (𝜓 ∧ 𝜒)) ▶ (𝜑 ↔ (𝜓 ∧ 𝜒)) ) | |
2 | biimpr 209 | . . . . 5 ⊢ ((𝜑 ↔ (𝜓 ∧ 𝜒)) → ((𝜓 ∧ 𝜒) → 𝜑)) | |
3 | 1, 2 | e1a 37873 | . . . 4 ⊢ ( (𝜑 ↔ (𝜓 ∧ 𝜒)) ▶ ((𝜓 ∧ 𝜒) → 𝜑) ) |
4 | pm3.3 459 | . . . 4 ⊢ (((𝜓 ∧ 𝜒) → 𝜑) → (𝜓 → (𝜒 → 𝜑))) | |
5 | 3, 4 | e1a 37873 | . . 3 ⊢ ( (𝜑 ↔ (𝜓 ∧ 𝜒)) ▶ (𝜓 → (𝜒 → 𝜑)) ) |
6 | pm2.04 88 | . . 3 ⊢ ((𝜓 → (𝜒 → 𝜑)) → (𝜒 → (𝜓 → 𝜑))) | |
7 | 5, 6 | e1a 37873 | . 2 ⊢ ( (𝜑 ↔ (𝜓 ∧ 𝜒)) ▶ (𝜒 → (𝜓 → 𝜑)) ) |
8 | 7 | in1 37808 | 1 ⊢ ((𝜑 ↔ (𝜓 ∧ 𝜒)) → (𝜒 → (𝜓 → 𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 |
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 196 df-an 385 df-vd1 37807 |
This theorem is referenced by: (None) |
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