![]() |
Mathbox for Alan Sare |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > con3ALTVD | Structured version Visualization version GIF version |
Description: The following User's Proof is a Natural Deduction Sequent Calculus
transcription of the Fitch-style Natural Deduction proof of Theorem 7 of
Section 14 of [Margaris] p. 60 ( which is con3 149). The same proof
may also be interpreted to be a Virtual Deduction Hilbert-style
axiomatic proof. It was completed automatically by the tools program
completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm
Megill's Metamath Proof Assistant. con3ALT2 39053 is con3ALTVD 39466
without virtual deductions and was automatically derived
from con3ALTVD 39466. Step i of the User's Proof corresponds to
step i of the Fitch-style proof.
|
Ref | Expression |
---|---|
con3ALTVD | ⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idn1 39107 | . . . . . 6 ⊢ ( (𝜑 → 𝜓) ▶ (𝜑 → 𝜓) ) | |
2 | idn2 39155 | . . . . . . 7 ⊢ ( (𝜑 → 𝜓) , ¬ ¬ 𝜑 ▶ ¬ ¬ 𝜑 ) | |
3 | notnotr 125 | . . . . . . 7 ⊢ (¬ ¬ 𝜑 → 𝜑) | |
4 | 2, 3 | e2 39173 | . . . . . 6 ⊢ ( (𝜑 → 𝜓) , ¬ ¬ 𝜑 ▶ 𝜑 ) |
5 | id 22 | . . . . . 6 ⊢ ((𝜑 → 𝜓) → (𝜑 → 𝜓)) | |
6 | 1, 4, 5 | e12 39268 | . . . . 5 ⊢ ( (𝜑 → 𝜓) , ¬ ¬ 𝜑 ▶ 𝜓 ) |
7 | notnot 136 | . . . . 5 ⊢ (𝜓 → ¬ ¬ 𝜓) | |
8 | 6, 7 | e2 39173 | . . . 4 ⊢ ( (𝜑 → 𝜓) , ¬ ¬ 𝜑 ▶ ¬ ¬ 𝜓 ) |
9 | 8 | in2 39147 | . . 3 ⊢ ( (𝜑 → 𝜓) ▶ (¬ ¬ 𝜑 → ¬ ¬ 𝜓) ) |
10 | con4 112 | . . 3 ⊢ ((¬ ¬ 𝜑 → ¬ ¬ 𝜓) → (¬ 𝜓 → ¬ 𝜑)) | |
11 | 9, 10 | e1a 39169 | . 2 ⊢ ( (𝜑 → 𝜓) ▶ (¬ 𝜓 → ¬ 𝜑) ) |
12 | 11 | in1 39104 | 1 ⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 |
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 197 df-an 385 df-vd1 39103 df-vd2 39111 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |