Metamath Proof Explorer

Theorem ee33VD

Description: Non-virtual deduction form of e33 . 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. ee33 is ee33VD without virtual deductions and was automatically derived from ee33VD .

h1:: |- ( ph -> ( ps -> ( ch -> th ) ) )
h2:: |- ( ph -> ( ps -> ( ch -> ta ) ) )
h3:: |- ( th -> ( ta -> et ) )
4:1,3: |- ( ph -> ( ps -> ( ch -> ( ta -> et ) ) ) )
5:4: |- ( ta -> ( ph -> ( ps -> ( ch -> et ) ) ) )
6:2,5: |- ( ph -> ( ps -> ( ch -> ( ph -> ( ps -> ( ch -> et ) ) ) ) ) )
7:6: |- ( ps -> ( ch -> ( ph -> ( ps -> ( ch -> et ) ) ) ) )
8:7: |- ( ch -> ( ph -> ( ps -> ( ch -> et ) ) ) )
qed:8: |- ( ph -> ( ps -> ( ch -> et ) ) )
(Contributed by Alan Sare, 18-Mar-2012) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypotheses ee33VD.1 ⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) )
ee33VD.2 ⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜏 ) ) )
ee33VD.3 ⊢ ( 𝜃 → ( 𝜏 → 𝜂 ) )
Assertion ee33VD ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜂 ) ) )

Proof

Step Hyp Ref Expression
1 ee33VD.1 ⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) )
2 ee33VD.2 ⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜏 ) ) )
3 ee33VD.3 ⊢ ( 𝜃 → ( 𝜏 → 𝜂 ) )
4 1 3 syl8 ⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → ( 𝜏 → 𝜂 ) ) ) )
5 4 com4r ⊢ ( 𝜏 → ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜂 ) ) ) )
6 2 5 syl8 ⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜂 ) ) ) ) ) )
7 pm2.43cbi ⊢ ( ( 𝜑 → ( 𝜓 → ( 𝜒 → ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜂 ) ) ) ) ) ) ↔ ( 𝜓 → ( 𝜒 → ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜂 ) ) ) ) ) )
8 7 biimpi ⊢ ( ( 𝜑 → ( 𝜓 → ( 𝜒 → ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜂 ) ) ) ) ) ) → ( 𝜓 → ( 𝜒 → ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜂 ) ) ) ) ) )
9 6 8 e0a ⊢ ( 𝜓 → ( 𝜒 → ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜂 ) ) ) ) )
10 pm2.43cbi ⊢ ( ( 𝜓 → ( 𝜒 → ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜂 ) ) ) ) ) ↔ ( 𝜒 → ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜂 ) ) ) ) )
11 10 biimpi ⊢ ( ( 𝜓 → ( 𝜒 → ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜂 ) ) ) ) ) → ( 𝜒 → ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜂 ) ) ) ) )
12 9 11 e0a ⊢ ( 𝜒 → ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜂 ) ) ) )
13 pm2.43cbi ⊢ ( ( 𝜒 → ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜂 ) ) ) ) ↔ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜂 ) ) ) )
14 13 biimpi ⊢ ( ( 𝜒 → ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜂 ) ) ) ) → ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜂 ) ) ) )
15 12 14 e0a ⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜂 ) ) )