Theorem ee222 44465

Description: e222 44599 without virtual deduction connectives. Special theorem needed for the Virtual Deduction translation tool. (Contributed by Alan Sare, 7-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
ee222.1	⊢ (𝜑 → (𝜓 → 𝜒))
ee222.2	⊢ (𝜑 → (𝜓 → 𝜃))
ee222.3	⊢ (𝜑 → (𝜓 → 𝜏))
ee222.4	⊢ (𝜒 → (𝜃 → (𝜏 → 𝜂)))

Assertion

Ref	Expression
ee222	⊢ (𝜑 → (𝜓 → 𝜂))

Proof of Theorem ee222

Step	Hyp	Ref	Expression
1		ee222.1	. . . 4 ⊢ (𝜑 → (𝜓 → 𝜒))
2	1	imp 406	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
3		ee222.2	. . . 4 ⊢ (𝜑 → (𝜓 → 𝜃))
4	3	imp 406	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜃)
5		ee222.3	. . . 4 ⊢ (𝜑 → (𝜓 → 𝜏))
6	5	imp 406	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜏)
7		ee222.4	. . 3 ⊢ (𝜒 → (𝜃 → (𝜏 → 𝜂)))
8	2, 4, 6, 7	syl3c 66	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝜂)
9	8	ex 412	1 ⊢ (𝜑 → (𝜓 → 𝜂))

Syntax hints:   → wi 4   ∧ 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

This theorem is referenced by:  ee121  44468  ee122  44469  tratrb  44499  ee220  44601  ee202  44603  ee022  44605  ee002  44607  ee020  44609  ee200  44611  ee221  44613  ee212  44615  ee112  44618  ee211  44621  ee210  44623  ee201  44625  ee120  44627  ee021  44629  ee012  44631  ee102  44633  suctrALT2  44799