Theorem exbiriVD 41926

Description: Virtual deduction proof of exbiri 811. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.

h1::	⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))
2::	⊢ (   𝜑   ▶   𝜑   )
3::	⊢ (   𝜑   ,   𝜓   ▶   𝜓   )
4::	⊢ (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜃   )
5:2,1,?: e10 41766	⊢ (   𝜑   ▶   (𝜓 → (𝜒 ↔ 𝜃))   )
6:3,5,?: e21 41802	⊢ (   𝜑   ,   𝜓   ▶   (𝜒 ↔ 𝜃)   )
7:4,6,?: e32 41830	⊢ (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜒   )
8:7:	⊢ (   𝜑   ,   𝜓   ▶   (𝜃 → 𝜒)   )
9:8:	⊢ (   𝜑   ▶   (𝜓 → (𝜃 → 𝜒))   )
qed:9:	⊢ (𝜑 → (𝜓 → (𝜃 → 𝜒)))

(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
exbiriVD.1	⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))

Assertion

Ref	Expression
exbiriVD	⊢ (𝜑 → (𝜓 → (𝜃 → 𝜒)))

Proof of Theorem exbiriVD

Step	Hyp	Ref	Expression
1		idn3 41687	. . . . 5 ⊢ (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜃   )
2		idn2 41685	. . . . . 6 ⊢ (   𝜑   ,   𝜓   ▶   𝜓   )
3		idn1 41646	. . . . . . 7 ⊢ (   𝜑   ▶   𝜑   )
4		exbiriVD.1	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))
5		pm3.3 453	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) → (𝜑 → (𝜓 → (𝜒 ↔ 𝜃))))
6	5	com12 32	. . . . . . 7 ⊢ (𝜑 → (((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) → (𝜓 → (𝜒 ↔ 𝜃))))
7	3, 4, 6	e10 41766	. . . . . 6 ⊢ (   𝜑   ▶   (𝜓 → (𝜒 ↔ 𝜃))   )
8		pm2.27 42	. . . . . 6 ⊢ (𝜓 → ((𝜓 → (𝜒 ↔ 𝜃)) → (𝜒 ↔ 𝜃)))
9	2, 7, 8	e21 41802	. . . . 5 ⊢ (   𝜑   ,   𝜓   ▶   (𝜒 ↔ 𝜃)   )
10		biimpr 223	. . . . . 6 ⊢ ((𝜒 ↔ 𝜃) → (𝜃 → 𝜒))
11	10	com12 32	. . . . 5 ⊢ (𝜃 → ((𝜒 ↔ 𝜃) → 𝜒))
12	1, 9, 11	e32 41830	. . . 4 ⊢ (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜒   )
13	12	in3 41681	. . 3 ⊢ (   𝜑   ,   𝜓   ▶   (𝜃 → 𝜒)   )
14	13	in2 41677	. 2 ⊢ (   𝜑   ▶   (𝜓 → (𝜃 → 𝜒))   )
15	14	in1 41643	1 ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜒)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400

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 210  df-an 401  df-3an 1087  df-vd1 41642  df-vd2 41650  df-vd3 41662

This theorem is referenced by: (None)