Theorem con5VD 46142

Description: Virtual deduction proof of con5 45765. 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 45765 is con5VD 46142 without virtual deductions and was automatically derived from con5VD 46142.

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 45817	. . . . 5 ⊢ (   (𝜑 ↔ ¬ 𝜓)   ▶   (𝜑 ↔ ¬ 𝜓)   )
2		biimpr 223	. . . . 5 ⊢ ((𝜑 ↔ ¬ 𝜓) → (¬ 𝜓 → 𝜑))
3	1, 2	e1a 45870	. . . 4 ⊢ (   (𝜑 ↔ ¬ 𝜓)   ▶   (¬ 𝜓 → 𝜑)   )
4		con3 154	. . . 4 ⊢ ((¬ 𝜓 → 𝜑) → (¬ 𝜑 → ¬ ¬ 𝜓))
5	3, 4	e1a 45870	. . 3 ⊢ (   (𝜑 ↔ ¬ 𝜓)   ▶   (¬ 𝜑 → ¬ ¬ 𝜓)   )
6		notnotb 318	. . 3 ⊢ (𝜓 ↔ ¬ ¬ 𝜓)
7		imbi2 351	. . . 4 ⊢ ((𝜓 ↔ ¬ ¬ 𝜓) → ((¬ 𝜑 → 𝜓) ↔ (¬ 𝜑 → ¬ ¬ 𝜓)))
8	7	biimprcd 253	. . 3 ⊢ ((¬ 𝜑 → ¬ ¬ 𝜓) → ((𝜓 ↔ ¬ ¬ 𝜓) → (¬ 𝜑 → 𝜓)))
9	5, 6, 8	e10 45937	. 2 ⊢ (   (𝜑 ↔ ¬ 𝜓)   ▶   (¬ 𝜑 → 𝜓)   )
10	9	in1 45814	1 ⊢ ((𝜑 ↔ ¬ 𝜓) → (¬ 𝜑 → 𝜓))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-vd1 45813

This theorem is used by: (None)