Theorem exbirVD 42473

Description: Virtual deduction proof of exbir 42098. 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.

1::	⊢ (   ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))    ▶   ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))   )
2::	⊢ (   ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))   ,    (𝜑 ∧ 𝜓)   ▶   (𝜑 ∧ 𝜓)   )
3::	⊢ (   ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))   ,    (𝜑 ∧ 𝜓), 𝜃   ▶   𝜃   )
5:1,2,?: e12 42344	⊢ (   ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)), (𝜑 ∧ 𝜓)   ▶   (𝜒 ↔ 𝜃)   )
6:3,5,?: e32 42378	⊢ (   ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)), (𝜑 ∧ 𝜓), 𝜃   ▶   𝜒   )
7:6:	⊢ (   ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)), (𝜑 ∧ 𝜓)   ▶   (𝜃 → 𝜒)   )
8:7:	⊢ (   ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))    ▶   ((𝜑 ∧ 𝜓) → (𝜃 → 𝜒))   )
9:8,?: e1a 42247	⊢ (   ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))   ▶   (𝜑 → (𝜓 → (𝜃 → 𝜒)))   )
qed:9:	⊢ (((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) → (𝜑 → (𝜓 → (𝜃 → 𝜒))))

(Contributed by Alan Sare, 13-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
exbirVD	⊢ (((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) → (𝜑 → (𝜓 → (𝜃 → 𝜒))))

Proof of Theorem exbirVD

Step	Hyp	Ref	Expression
1		idn3 42235	. . . . . 6 ⊢ (   ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))   ,   (𝜑 ∧ 𝜓)   ,   𝜃   ▶   𝜃   )
2		idn1 42194	. . . . . . 7 ⊢ (   ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))   ▶   ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))   )
3		idn2 42233	. . . . . . 7 ⊢ (   ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))   ,   (𝜑 ∧ 𝜓)   ▶   (𝜑 ∧ 𝜓)   )
4		id 22	. . . . . . 7 ⊢ (((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) → ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)))
5	2, 3, 4	e12 42344	. . . . . 6 ⊢ (   ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))   ,   (𝜑 ∧ 𝜓)   ▶   (𝜒 ↔ 𝜃)   )
6		biimpr 219	. . . . . . 7 ⊢ ((𝜒 ↔ 𝜃) → (𝜃 → 𝜒))
7	6	com12 32	. . . . . 6 ⊢ (𝜃 → ((𝜒 ↔ 𝜃) → 𝜒))
8	1, 5, 7	e32 42378	. . . . 5 ⊢ (   ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))   ,   (𝜑 ∧ 𝜓)   ,   𝜃   ▶   𝜒   )
9	8	in3 42229	. . . 4 ⊢ (   ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))   ,   (𝜑 ∧ 𝜓)   ▶   (𝜃 → 𝜒)   )
10	9	in2 42225	. . 3 ⊢ (   ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))   ▶   ((𝜑 ∧ 𝜓) → (𝜃 → 𝜒))   )
11		pm3.3 449	. . 3 ⊢ (((𝜑 ∧ 𝜓) → (𝜃 → 𝜒)) → (𝜑 → (𝜓 → (𝜃 → 𝜒))))
12	10, 11	e1a 42247	. 2 ⊢ (   ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))   ▶   (𝜑 → (𝜓 → (𝜃 → 𝜒)))   )
13	12	in1 42191	1 ⊢ (((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) → (𝜑 → (𝜓 → (𝜃 → 𝜒))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396

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 206  df-an 397  df-3an 1088  df-vd1 42190  df-vd2 42198  df-vd3 42210

This theorem is referenced by: (None)