Theorem exbiriVD 41560

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 41400	⊢ (   𝜑   ▶   (𝜓 → (𝜒 ↔ 𝜃))   )
6:3,5,?: e21 41436	⊢ (   𝜑   ,   𝜓   ▶   (𝜒 ↔ 𝜃)   )
7:4,6,?: e32 41464	⊢ (   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 41321	. . . . 5 ⊢ (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜃   )
2		idn2 41319	. . . . . 6 ⊢ (   𝜑   ,   𝜓   ▶   𝜓   )
3		idn1 41280	. . . . . . 7 ⊢ (   𝜑   ▶   𝜑   )
4		exbiriVD.1	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))
5		pm3.3 452	. . . . . . . 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 41400	. . . . . 6 ⊢ (   𝜑   ▶   (𝜓 → (𝜒 ↔ 𝜃))   )
8		pm2.27 42	. . . . . 6 ⊢ (𝜓 → ((𝜓 → (𝜒 ↔ 𝜃)) → (𝜒 ↔ 𝜃)))
9	2, 7, 8	e21 41436	. . . . 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 41464	. . . 4 ⊢ (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜒   )
13	12	in3 41315	. . 3 ⊢ (   𝜑   ,   𝜓   ▶   (𝜃 → 𝜒)   )
14	13	in2 41311	. 2 ⊢ (   𝜑   ▶   (𝜓 → (𝜃 → 𝜒))   )
15	14	in1 41277	1 ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜒)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399

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 400  df-3an 1086  df-vd1 41276  df-vd2 41284  df-vd3 41296

This theorem is referenced by: (None)