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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 148). 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 37757 is con3ALTVD 38174
without virtual deductions and was automatically derived
from con3ALTVD 38174. 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 37811 | . . . . . 6 ⊢ ( (𝜑 → 𝜓) ▶ (𝜑 → 𝜓) ) | |
2 | idn2 37859 | . . . . . . 7 ⊢ ( (𝜑 → 𝜓) , ¬ ¬ 𝜑 ▶ ¬ ¬ 𝜑 ) | |
3 | notnotr 124 | . . . . . . 7 ⊢ (¬ ¬ 𝜑 → 𝜑) | |
4 | 2, 3 | e2 37877 | . . . . . 6 ⊢ ( (𝜑 → 𝜓) , ¬ ¬ 𝜑 ▶ 𝜑 ) |
5 | id 22 | . . . . . 6 ⊢ ((𝜑 → 𝜓) → (𝜑 → 𝜓)) | |
6 | 1, 4, 5 | e12 37972 | . . . . 5 ⊢ ( (𝜑 → 𝜓) , ¬ ¬ 𝜑 ▶ 𝜓 ) |
7 | notnot 135 | . . . . 5 ⊢ (𝜓 → ¬ ¬ 𝜓) | |
8 | 6, 7 | e2 37877 | . . . 4 ⊢ ( (𝜑 → 𝜓) , ¬ ¬ 𝜑 ▶ ¬ ¬ 𝜓 ) |
9 | 8 | in2 37851 | . . 3 ⊢ ( (𝜑 → 𝜓) ▶ (¬ ¬ 𝜑 → ¬ ¬ 𝜓) ) |
10 | con4 111 | . . 3 ⊢ ((¬ ¬ 𝜑 → ¬ ¬ 𝜓) → (¬ 𝜓 → ¬ 𝜑)) | |
11 | 9, 10 | e1a 37873 | . 2 ⊢ ( (𝜑 → 𝜓) ▶ (¬ 𝜓 → ¬ 𝜑) ) |
12 | 11 | in1 37808 | 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 196 df-an 385 df-vd1 37807 df-vd2 37815 |
This theorem is referenced by: (None) |
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