Theorem eel2122old 42338

Description: el2122old 42339 without virtual deductions. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
eel2122old.1	⊢ ((𝜑 ∧ 𝜓) → 𝜒)
eel2122old.2	⊢ (𝜓 → 𝜃)
eel2122old.3	⊢ (𝜓 → 𝜏)
eel2122old.4	⊢ ((𝜒 ∧ 𝜃 ∧ 𝜏) → 𝜂)

Assertion

Ref	Expression
eel2122old	⊢ ((𝜑 ∧ 𝜓) → 𝜂)

Proof of Theorem eel2122old

Step	Hyp	Ref	Expression
1		eel2122old.3	. . . . . 6 ⊢ (𝜓 → 𝜏)
2		eel2122old.2	. . . . . . 7 ⊢ (𝜓 → 𝜃)
3		eel2122old.1	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
4		eel2122old.4	. . . . . . . . 9 ⊢ ((𝜒 ∧ 𝜃 ∧ 𝜏) → 𝜂)
5	4	3exp 1118	. . . . . . . 8 ⊢ (𝜒 → (𝜃 → (𝜏 → 𝜂)))
6	3, 5	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → (𝜃 → (𝜏 → 𝜂)))
7	2, 6	syl5 34	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → (𝜓 → (𝜏 → 𝜂)))
8	1, 7	syl7 74	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝜓 → (𝜓 → 𝜂)))
9	8	ex 413	. . . 4 ⊢ (𝜑 → (𝜓 → (𝜓 → (𝜓 → 𝜂))))
10	9	pm2.43d 53	. . 3 ⊢ (𝜑 → (𝜓 → (𝜓 → 𝜂)))
11	10	pm2.43d 53	. 2 ⊢ (𝜑 → (𝜓 → 𝜂))
12	11	imp 407	1 ⊢ ((𝜑 ∧ 𝜓) → 𝜂)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086

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  df-3an 1088

This theorem is referenced by:  el2122old  42339  suctrALTcf  42542