Theorem ee222 42011

Description: e222 42145 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 206  df-an 396

This theorem is referenced by:  ee121  42014  ee122  42015  tratrb  42045  ee220  42147  ee202  42149  ee022  42151  ee002  42153  ee020  42155  ee200  42157  ee221  42159  ee212  42161  ee112  42164  ee211  42167  ee210  42169  ee201  42171  ee120  42173  ee021  42175  ee012  42177  ee102  42179  suctrALT2  42346