Theorem ee222 44492

Description: e222 44626 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  44495  ee122  44496  tratrb  44526  ee220  44628  ee202  44630  ee022  44632  ee002  44634  ee020  44636  ee200  44638  ee221  44640  ee212  44642  ee112  44645  ee211  44648  ee210  44650  ee201  44652  ee120  44654  ee021  44656  ee012  44658  ee102  44660  suctrALT2  44826