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Theorem simplbi2comtVD 39873
Description: Virtual deduction proof of simplbi2comt 496. 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 496 is simplbi2comtVD 39873 without virtual deductions and was automatically derived from simplbi2comtVD 39873.
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 39549 . . . . 5 (   (𝜑 ↔ (𝜓𝜒))   ▶   (𝜑 ↔ (𝜓𝜒))   )
2 biimpr 212 . . . . 5 ((𝜑 ↔ (𝜓𝜒)) → ((𝜓𝜒) → 𝜑))
31, 2e1a 39611 . . . 4 (   (𝜑 ↔ (𝜓𝜒))   ▶   ((𝜓𝜒) → 𝜑)   )
4 pm3.3 440 . . . 4 (((𝜓𝜒) → 𝜑) → (𝜓 → (𝜒𝜑)))
53, 4e1a 39611 . . 3 (   (𝜑 ↔ (𝜓𝜒))   ▶   (𝜓 → (𝜒𝜑))   )
6 pm2.04 90 . . 3 ((𝜓 → (𝜒𝜑)) → (𝜒 → (𝜓𝜑)))
75, 6e1a 39611 . 2 (   (𝜑 ↔ (𝜓𝜒))   ▶   (𝜒 → (𝜓𝜑))   )
87in1 39546 1 ((𝜑 ↔ (𝜓𝜒)) → (𝜒 → (𝜓𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385
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 199  df-an 386  df-vd1 39545
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator