Theorem syl5impVD 42372

Description: Virtual deduction proof of syl5imp 42021. 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))   ▶   (𝜑 → (𝜓 → 𝜒))   )
2:1,?: e1a 42136	⊢ (   (𝜑 → (𝜓 → 𝜒))   ▶   (𝜓 → (𝜑 → 𝜒))   )
3::	⊢ (   (𝜑 → (𝜓 → 𝜒))   ,   (𝜃 → 𝜓)   ▶   (𝜃 → 𝜓)   )
4:3,2,?: e21 42239	⊢ (   (𝜑 → (𝜓 → 𝜒))   ,   (𝜃 → 𝜓)   ▶   (𝜃 → (𝜑 → 𝜒))   )
5:4,?: e2 42140	⊢ (   (𝜑 → (𝜓 → 𝜒))   ,   (𝜃 → 𝜓)   ▶   (𝜑 → (𝜃 → 𝜒))   )
6:5:	⊢ (   (𝜑 → (𝜓 → 𝜒))   ▶   ((𝜃 → 𝜓) → (𝜑 → (𝜃 → 𝜒)))   )
qed:6:	⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜃 → 𝜓) → (𝜑 → (𝜃 → 𝜒))))

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

Assertion

Ref	Expression
syl5impVD	⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜃 → 𝜓) → (𝜑 → (𝜃 → 𝜒))))

Proof of Theorem syl5impVD

Step	Hyp	Ref	Expression
1		idn2 42122	. . . . 5 ⊢ (   (𝜑 → (𝜓 → 𝜒))   ,   (𝜃 → 𝜓)   ▶   (𝜃 → 𝜓)   )
2		idn1 42083	. . . . . 6 ⊢ (   (𝜑 → (𝜓 → 𝜒))   ▶   (𝜑 → (𝜓 → 𝜒))   )
3		pm2.04 90	. . . . . 6 ⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒)))
4	2, 3	e1a 42136	. . . . 5 ⊢ (   (𝜑 → (𝜓 → 𝜒))   ▶   (𝜓 → (𝜑 → 𝜒))   )
5		imim1 83	. . . . 5 ⊢ ((𝜃 → 𝜓) → ((𝜓 → (𝜑 → 𝜒)) → (𝜃 → (𝜑 → 𝜒))))
6	1, 4, 5	e21 42239	. . . 4 ⊢ (   (𝜑 → (𝜓 → 𝜒))   ,   (𝜃 → 𝜓)   ▶   (𝜃 → (𝜑 → 𝜒))   )
7		pm2.04 90	. . . 4 ⊢ ((𝜃 → (𝜑 → 𝜒)) → (𝜑 → (𝜃 → 𝜒)))
8	6, 7	e2 42140	. . 3 ⊢ (   (𝜑 → (𝜓 → 𝜒))   ,   (𝜃 → 𝜓)   ▶   (𝜑 → (𝜃 → 𝜒))   )
9	8	in2 42114	. 2 ⊢ (   (𝜑 → (𝜓 → 𝜒))   ▶   ((𝜃 → 𝜓) → (𝜑 → (𝜃 → 𝜒)))   )
10	9	in1 42080	1 ⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜃 → 𝜓) → (𝜑 → (𝜃 → 𝜒))))

Syntax hints:   → wi 4

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 396  df-vd1 42079  df-vd2 42087

This theorem is referenced by: (None)