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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 38736 is con3ALTVD 39152
without virtual deductions and was automatically derived
from con3ALTVD 39152. 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 38790 | . . . . . 6 ⊢ ( (𝜑 → 𝜓) ▶ (𝜑 → 𝜓) ) | |
2 | idn2 38838 | . . . . . . 7 ⊢ ( (𝜑 → 𝜓) , ¬ ¬ 𝜑 ▶ ¬ ¬ 𝜑 ) | |
3 | notnotr 125 | . . . . . . 7 ⊢ (¬ ¬ 𝜑 → 𝜑) | |
4 | 2, 3 | e2 38856 | . . . . . 6 ⊢ ( (𝜑 → 𝜓) , ¬ ¬ 𝜑 ▶ 𝜑 ) |
5 | id 22 | . . . . . 6 ⊢ ((𝜑 → 𝜓) → (𝜑 → 𝜓)) | |
6 | 1, 4, 5 | e12 38951 | . . . . 5 ⊢ ( (𝜑 → 𝜓) , ¬ ¬ 𝜑 ▶ 𝜓 ) |
7 | notnot 136 | . . . . 5 ⊢ (𝜓 → ¬ ¬ 𝜓) | |
8 | 6, 7 | e2 38856 | . . . 4 ⊢ ( (𝜑 → 𝜓) , ¬ ¬ 𝜑 ▶ ¬ ¬ 𝜓 ) |
9 | 8 | in2 38830 | . . 3 ⊢ ( (𝜑 → 𝜓) ▶ (¬ ¬ 𝜑 → ¬ ¬ 𝜓) ) |
10 | con4 112 | . . 3 ⊢ ((¬ ¬ 𝜑 → ¬ ¬ 𝜓) → (¬ 𝜓 → ¬ 𝜑)) | |
11 | 9, 10 | e1a 38852 | . 2 ⊢ ( (𝜑 → 𝜓) ▶ (¬ 𝜓 → ¬ 𝜑) ) |
12 | 11 | in1 38787 | 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 386 df-vd1 38786 df-vd2 38794 |
This theorem is referenced by: (None) |
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