Theorem con5VD 44889

Description: Virtual deduction proof of con5 44512. 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 44512 is con5VD 44889 without virtual deductions and was automatically derived from con5VD 44889.

1::	⊢ (   (𝜑 ↔ ¬ 𝜓)   ▶   (𝜑 ↔ ¬ 𝜓)   )
2:1:	⊢ (   (𝜑 ↔ ¬ 𝜓)   ▶   (¬ 𝜓 → 𝜑)   )
3:2:	⊢ (   (𝜑 ↔ ¬ 𝜓)   ▶   (¬ 𝜑 → ¬ ¬ 𝜓 )   )
4::	⊢ (𝜓 ↔ ¬ ¬ 𝜓)
5:3,4:	⊢ (   (𝜑 ↔ ¬ 𝜓)   ▶   (¬ 𝜑 → 𝜓)   )
qed:5:	⊢ ((𝜑 ↔ ¬ 𝜓) → (¬ 𝜑 → 𝜓))

(Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
con5VD	⊢ ((𝜑 ↔ ¬ 𝜓) → (¬ 𝜑 → 𝜓))

Proof of Theorem con5VD

Step	Hyp	Ref	Expression
1		idn1 44564	. . . . 5 ⊢ (   (𝜑 ↔ ¬ 𝜓)   ▶   (𝜑 ↔ ¬ 𝜓)   )
2		biimpr 220	. . . . 5 ⊢ ((𝜑 ↔ ¬ 𝜓) → (¬ 𝜓 → 𝜑))
3	1, 2	e1a 44617	. . . 4 ⊢ (   (𝜑 ↔ ¬ 𝜓)   ▶   (¬ 𝜓 → 𝜑)   )
4		con3 153	. . . 4 ⊢ ((¬ 𝜓 → 𝜑) → (¬ 𝜑 → ¬ ¬ 𝜓))
5	3, 4	e1a 44617	. . 3 ⊢ (   (𝜑 ↔ ¬ 𝜓)   ▶   (¬ 𝜑 → ¬ ¬ 𝜓)   )
6		notnotb 315	. . 3 ⊢ (𝜓 ↔ ¬ ¬ 𝜓)
7		imbi2 348	. . . 4 ⊢ ((𝜓 ↔ ¬ ¬ 𝜓) → ((¬ 𝜑 → 𝜓) ↔ (¬ 𝜑 → ¬ ¬ 𝜓)))
8	7	biimprcd 250	. . 3 ⊢ ((¬ 𝜑 → ¬ ¬ 𝜓) → ((𝜓 ↔ ¬ ¬ 𝜓) → (¬ 𝜑 → 𝜓)))
9	5, 6, 8	e10 44684	. 2 ⊢ (   (𝜑 ↔ ¬ 𝜓)   ▶   (¬ 𝜑 → 𝜓)   )
10	9	in1 44561	1 ⊢ ((𝜑 ↔ ¬ 𝜓) → (¬ 𝜑 → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206

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 207  df-vd1 44560

This theorem is referenced by: (None)