Theorem con5VD 45435

Description: Virtual deduction proof of con5 45058. 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 45058 is con5VD 45435 without virtual deductions and was automatically derived from con5VD 45435.

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 45110	. . . . 5 ⊢ (   (𝜑 ↔ ¬ 𝜓)   ▶   (𝜑 ↔ ¬ 𝜓)   )
2		biimpr 222	. . . . 5 ⊢ ((𝜑 ↔ ¬ 𝜓) → (¬ 𝜓 → 𝜑))
3	1, 2	e1a 45163	. . . 4 ⊢ (   (𝜑 ↔ ¬ 𝜓)   ▶   (¬ 𝜓 → 𝜑)   )
4		con3 153	. . . 4 ⊢ ((¬ 𝜓 → 𝜑) → (¬ 𝜑 → ¬ ¬ 𝜓))
5	3, 4	e1a 45163	. . 3 ⊢ (   (𝜑 ↔ ¬ 𝜓)   ▶   (¬ 𝜑 → ¬ ¬ 𝜓)   )
6		notnotb 317	. . 3 ⊢ (𝜓 ↔ ¬ ¬ 𝜓)
7		imbi2 350	. . . 4 ⊢ ((𝜓 ↔ ¬ ¬ 𝜓) → ((¬ 𝜑 → 𝜓) ↔ (¬ 𝜑 → ¬ ¬ 𝜓)))
8	7	biimprcd 252	. . 3 ⊢ ((¬ 𝜑 → ¬ ¬ 𝜓) → ((𝜓 ↔ ¬ ¬ 𝜓) → (¬ 𝜑 → 𝜓)))
9	5, 6, 8	e10 45230	. 2 ⊢ (   (𝜑 ↔ ¬ 𝜓)   ▶   (¬ 𝜑 → 𝜓)   )
10	9	in1 45107	1 ⊢ ((𝜑 ↔ ¬ 𝜓) → (¬ 𝜑 → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208

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 209  df-vd1 45106

This theorem is referenced by: (None)