Theorem exbiriVD 44878

Description: Virtual deduction proof of exbiri 810. 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 44719	⊢ (   𝜑   ▶   (𝜓 → (𝜒 ↔ 𝜃))   )
6:3,5,?: e21 44754	⊢ (   𝜑   ,   𝜓   ▶   (𝜒 ↔ 𝜃)   )
7:4,6,?: e32 44782	⊢ (   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 44640	. . . . 5 ⊢ (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜃   )
2		idn2 44638	. . . . . 6 ⊢ (   𝜑   ,   𝜓   ▶   𝜓   )
3		idn1 44599	. . . . . . 7 ⊢ (   𝜑   ▶   𝜑   )
4		exbiriVD.1	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))
5		pm3.3 448	. . . . . . . 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 44719	. . . . . 6 ⊢ (   𝜑   ▶   (𝜓 → (𝜒 ↔ 𝜃))   )
8		pm2.27 42	. . . . . 6 ⊢ (𝜓 → ((𝜓 → (𝜒 ↔ 𝜃)) → (𝜒 ↔ 𝜃)))
9	2, 7, 8	e21 44754	. . . . 5 ⊢ (   𝜑   ,   𝜓   ▶   (𝜒 ↔ 𝜃)   )
10		biimpr 220	. . . . . 6 ⊢ ((𝜒 ↔ 𝜃) → (𝜃 → 𝜒))
11	10	com12 32	. . . . 5 ⊢ (𝜃 → ((𝜒 ↔ 𝜃) → 𝜒))
12	1, 9, 11	e32 44782	. . . 4 ⊢ (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜒   )
13	12	in3 44634	. . 3 ⊢ (   𝜑   ,   𝜓   ▶   (𝜃 → 𝜒)   )
14	13	in2 44630	. 2 ⊢ (   𝜑   ▶   (𝜓 → (𝜃 → 𝜒))   )
15	14	in1 44596	1 ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜒)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395

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-an 396  df-3an 1088  df-vd1 44595  df-vd2 44603  df-vd3 44615

This theorem is referenced by: (None)