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Theorem simplbi2comtVD 41581
Description: Virtual deduction proof of simplbi2comt 505. 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. simplbi2comt 505 is simplbi2comtVD 41581 without virtual deductions and was automatically derived from simplbi2comtVD 41581.
1:: (   (𝜑 ↔ (𝜓𝜒))   ▶   (𝜑 ↔ ( 𝜓𝜒))   )
2:1: (   (𝜑 ↔ (𝜓𝜒))   ▶   ((𝜓𝜒 ) → 𝜑)   )
3:2: (   (𝜑 ↔ (𝜓𝜒))   ▶   (𝜓 → (𝜒 𝜑))   )
4:3: (   (𝜑 ↔ (𝜓𝜒))   ▶   (𝜒 → (𝜓 𝜑))   )
qed:4: ((𝜑 ↔ (𝜓𝜒)) → (𝜒 → (𝜓 𝜑)))
(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
simplbi2comtVD ((𝜑 ↔ (𝜓𝜒)) → (𝜒 → (𝜓𝜑)))

Proof of Theorem simplbi2comtVD
StepHypRef Expression
1 idn1 41267 . . . . 5 (   (𝜑 ↔ (𝜓𝜒))   ▶   (𝜑 ↔ (𝜓𝜒))   )
2 biimpr 223 . . . . 5 ((𝜑 ↔ (𝜓𝜒)) → ((𝜓𝜒) → 𝜑))
31, 2e1a 41320 . . . 4 (   (𝜑 ↔ (𝜓𝜒))   ▶   ((𝜓𝜒) → 𝜑)   )
4 pm3.3 452 . . . 4 (((𝜓𝜒) → 𝜑) → (𝜓 → (𝜒𝜑)))
53, 4e1a 41320 . . 3 (   (𝜑 ↔ (𝜓𝜒))   ▶   (𝜓 → (𝜒𝜑))   )
6 pm2.04 90 . . 3 ((𝜓 → (𝜒𝜑)) → (𝜒 → (𝜓𝜑)))
75, 6e1a 41320 . 2 (   (𝜑 ↔ (𝜓𝜒))   ▶   (𝜒 → (𝜓𝜑))   )
87in1 41264 1 ((𝜑 ↔ (𝜓𝜒)) → (𝜒 → (𝜓𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399
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 210  df-an 400  df-vd1 41263
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator