Theorem ee222 42451

Description: e222 42585 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 407	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
3		ee222.2	. . . 4 ⊢ (𝜑 → (𝜓 → 𝜃))
4	3	imp 407	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜃)
5		ee222.3	. . . 4 ⊢ (𝜑 → (𝜓 → 𝜏))
6	5	imp 407	. . 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 413	1 ⊢ (𝜑 → (𝜓 → 𝜂))

Syntax hints:   → wi 4   ∧ wa 396

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 397

This theorem is referenced by:  ee121  42454  ee122  42455  tratrb  42485  ee220  42587  ee202  42589  ee022  42591  ee002  42593  ee020  42595  ee200  42597  ee221  42599  ee212  42601  ee112  42604  ee211  42607  ee210  42609  ee201  42611  ee120  42613  ee021  42615  ee012  42617  ee102  42619  suctrALT2  42786