Theorem ee222 44951

Description: e222 45085 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  44954  ee122  44955  tratrb  44985  ee220  45087  ee202  45089  ee022  45091  ee002  45093  ee020  45095  ee200  45097  ee221  45099  ee212  45101  ee112  45104  ee211  45107  ee210  45109  ee201  45111  ee120  45113  ee021  45115  ee012  45117  ee102  45119  suctrALT2  45285