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Theorem simplbi2comtVD 43953
Description: Virtual deduction proof of simplbi2comt 500. 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 500 is simplbi2comtVD 43953 without virtual deductions and was automatically derived from simplbi2comtVD 43953.
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 43639 . . . . 5 (   (𝜑 ↔ (𝜓𝜒))   ▶   (𝜑 ↔ (𝜓𝜒))   )
2 biimpr 219 . . . . 5 ((𝜑 ↔ (𝜓𝜒)) → ((𝜓𝜒) → 𝜑))
31, 2e1a 43692 . . . 4 (   (𝜑 ↔ (𝜓𝜒))   ▶   ((𝜓𝜒) → 𝜑)   )
4 pm3.3 447 . . . 4 (((𝜓𝜒) → 𝜑) → (𝜓 → (𝜒𝜑)))
53, 4e1a 43692 . . 3 (   (𝜑 ↔ (𝜓𝜒))   ▶   (𝜓 → (𝜒𝜑))   )
6 pm2.04 90 . . 3 ((𝜓 → (𝜒𝜑)) → (𝜒 → (𝜓𝜑)))
75, 6e1a 43692 . 2 (   (𝜑 ↔ (𝜓𝜒))   ▶   (𝜒 → (𝜓𝜑))   )
87in1 43636 1 ((𝜑 ↔ (𝜓𝜒)) → (𝜒 → (𝜓𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394
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 395  df-vd1 43635
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator