Theorem exbiriVD 45676

Description: Virtual deduction proof of exbiri 823. 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 45517	⊢ (   𝜑   ▶   (𝜓 → (𝜒 ↔ 𝜃))   )
6:3,5,?: e21 45552	⊢ (   𝜑   ,   𝜓   ▶   (𝜒 ↔ 𝜃)   )
7:4,6,?: e32 45580	⊢ (   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 45438	. . . . 5 ⊢ (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜃   )
2		idn2 45436	. . . . . 6 ⊢ (   𝜑   ,   𝜓   ▶   𝜓   )
3		idn1 45397	. . . . . . 7 ⊢ (   𝜑   ▶   𝜑   )
4		exbiriVD.1	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))
5		pm3.3 454	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) → (𝜑 → (𝜓 → (𝜒 ↔ 𝜃))))
6	5	com12 33	. . . . . . 7 ⊢ (𝜑 → (((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) → (𝜓 → (𝜒 ↔ 𝜃))))
7	3, 4, 6	e10 45517	. . . . . 6 ⊢ (   𝜑   ▶   (𝜓 → (𝜒 ↔ 𝜃))   )
8		pm2.27 43	. . . . . 6 ⊢ (𝜓 → ((𝜓 → (𝜒 ↔ 𝜃)) → (𝜒 ↔ 𝜃)))
9	2, 7, 8	e21 45552	. . . . 5 ⊢ (   𝜑   ,   𝜓   ▶   (𝜒 ↔ 𝜃)   )
10		biimpr 223	. . . . . 6 ⊢ ((𝜒 ↔ 𝜃) → (𝜃 → 𝜒))
11	10	com12 33	. . . . 5 ⊢ (𝜃 → ((𝜒 ↔ 𝜃) → 𝜒))
12	1, 9, 11	e32 45580	. . . 4 ⊢ (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜒   )
13	12	in3 45432	. . 3 ⊢ (   𝜑   ,   𝜓   ▶   (𝜃 → 𝜒)   )
14	13	in2 45428	. 2 ⊢ (   𝜑   ▶   (𝜓 → (𝜃 → 𝜒))   )
15	14	in1 45394	1 ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜒)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401

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-an 402  df-3an 1105  df-vd1 45393  df-vd2 45401  df-vd3 45413

This theorem is used by: (None)