Theorem ee1111 42025

Description: Non-virtual deduction form of e1111 42184. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The completed Virtual Deduction Proof (not shown) was minimized. The minimized proof is shown.

h1::	⊢ (𝜑 → 𝜓)
h2::	⊢ (𝜑 → 𝜒)
h3::	⊢ (𝜑 → 𝜃)
h4::	⊢ (𝜑 → 𝜏)
h5::	⊢ (𝜓 → (𝜒 → (𝜃 → (𝜏 → 𝜂))))
6:1,5:	⊢ (𝜑 → (𝜒 → (𝜃 → (𝜏 → 𝜂))))
7:6:	⊢ (𝜒 → (𝜑 → (𝜃 → (𝜏 → 𝜂))))
8:2,7:	⊢ (𝜑 → (𝜑 → (𝜃 → (𝜏 → 𝜂))))
9:8:	⊢ (𝜑 → (𝜃 → (𝜏 → 𝜂)))
10:9:	⊢ (𝜃 → (𝜑 → (𝜏 → 𝜂)))
11:3,10:	⊢ (𝜑 → (𝜑 → (𝜏 → 𝜂)))
12:11:	⊢ (𝜑 → (𝜏 → 𝜂))
13:12:	⊢ (𝜏 → (𝜑 → 𝜂))
14:4,13:	⊢ (𝜑 → (𝜑 → 𝜂))
qed:14:	⊢ (𝜑 → 𝜂)

Hypotheses

Ref	Expression
ee1111.1	⊢ (𝜑 → 𝜓)
ee1111.2	⊢ (𝜑 → 𝜒)
ee1111.3	⊢ (𝜑 → 𝜃)
ee1111.4	⊢ (𝜑 → 𝜏)
ee1111.5	⊢ (𝜓 → (𝜒 → (𝜃 → (𝜏 → 𝜂))))

Assertion

Ref	Expression
ee1111	⊢ (𝜑 → 𝜂)

Proof of Theorem ee1111

Step	Hyp	Ref	Expression
1		ee1111.4	. 2 ⊢ (𝜑 → 𝜏)
2		ee1111.1	. . 3 ⊢ (𝜑 → 𝜓)
3		ee1111.2	. . 3 ⊢ (𝜑 → 𝜒)
4		ee1111.3	. . 3 ⊢ (𝜑 → 𝜃)
5		ee1111.5	. . 3 ⊢ (𝜓 → (𝜒 → (𝜃 → (𝜏 → 𝜂))))
6	2, 3, 4, 5	syl3c 66	. 2 ⊢ (𝜑 → (𝜏 → 𝜂))
7	1, 6	mpd 15	1 ⊢ (𝜑 → 𝜂)

Syntax hints:   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7

This theorem is referenced by:  e1111  42184