Description: Virtual deduction proof of con5 . 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. con5 is con5VD without virtual deductions and was automatically derived from con5VD .
| 1:: | |- (. ( ph <-> -. ps ) ->. ( ph <-> -. ps ) ). |
| 2:1: | |- (. ( ph <-> -. ps ) ->. ( -. ps -> ph ) ). |
| 3:2: | |- (. ( ph <-> -. ps ) ->. ( -. ph -> -. -. ps ) ). |
| 4:: | |- ( ps <-> -. -. ps ) |
| 5:3,4: | |- (. ( ph <-> -. ps ) ->. ( -. ph -> ps ) ). |
| qed:5: | |- ( ( ph <-> -. ps ) -> ( -. ph -> ps ) ) |
| Ref | Expression | ||
|---|---|---|---|
| Assertion | con5VD | ⊢ ( ( 𝜑 ↔ ¬ 𝜓 ) → ( ¬ 𝜑 → 𝜓 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | idn1 | ⊢ ( ( 𝜑 ↔ ¬ 𝜓 ) ▶ ( 𝜑 ↔ ¬ 𝜓 ) ) | |
| 2 | biimpr | ⊢ ( ( 𝜑 ↔ ¬ 𝜓 ) → ( ¬ 𝜓 → 𝜑 ) ) | |
| 3 | 1 2 | e1a | ⊢ ( ( 𝜑 ↔ ¬ 𝜓 ) ▶ ( ¬ 𝜓 → 𝜑 ) ) |
| 4 | con3 | ⊢ ( ( ¬ 𝜓 → 𝜑 ) → ( ¬ 𝜑 → ¬ ¬ 𝜓 ) ) | |
| 5 | 3 4 | e1a | ⊢ ( ( 𝜑 ↔ ¬ 𝜓 ) ▶ ( ¬ 𝜑 → ¬ ¬ 𝜓 ) ) |
| 6 | notnotb | ⊢ ( 𝜓 ↔ ¬ ¬ 𝜓 ) | |
| 7 | imbi2 | ⊢ ( ( 𝜓 ↔ ¬ ¬ 𝜓 ) → ( ( ¬ 𝜑 → 𝜓 ) ↔ ( ¬ 𝜑 → ¬ ¬ 𝜓 ) ) ) | |
| 8 | 7 | biimprcd | ⊢ ( ( ¬ 𝜑 → ¬ ¬ 𝜓 ) → ( ( 𝜓 ↔ ¬ ¬ 𝜓 ) → ( ¬ 𝜑 → 𝜓 ) ) ) |
| 9 | 5 6 8 | e10 | ⊢ ( ( 𝜑 ↔ ¬ 𝜓 ) ▶ ( ¬ 𝜑 → 𝜓 ) ) |
| 10 | 9 | in1 | ⊢ ( ( 𝜑 ↔ ¬ 𝜓 ) → ( ¬ 𝜑 → 𝜓 ) ) |