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Theorem simplbi2comtVD 39438
Description: Virtual deduction proof of simplbi2comt 655. 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 655 is simplbi2comtVD 39438 without virtual deductions and was automatically derived from simplbi2comtVD 39438.
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 39107 . . . . 5 (   (𝜑 ↔ (𝜓𝜒))   ▶   (𝜑 ↔ (𝜓𝜒))   )
2 biimpr 210 . . . . 5 ((𝜑 ↔ (𝜓𝜒)) → ((𝜓𝜒) → 𝜑))
31, 2e1a 39169 . . . 4 (   (𝜑 ↔ (𝜓𝜒))   ▶   ((𝜓𝜒) → 𝜑)   )
4 pm3.3 459 . . . 4 (((𝜓𝜒) → 𝜑) → (𝜓 → (𝜒𝜑)))
53, 4e1a 39169 . . 3 (   (𝜑 ↔ (𝜓𝜒))   ▶   (𝜓 → (𝜒𝜑))   )
6 pm2.04 90 . . 3 ((𝜓 → (𝜒𝜑)) → (𝜒 → (𝜓𝜑)))
75, 6e1a 39169 . 2 (   (𝜑 ↔ (𝜓𝜒))   ▶   (𝜒 → (𝜓𝜑))   )
87in1 39104 1 ((𝜑 ↔ (𝜓𝜒)) → (𝜒 → (𝜓𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383
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 197  df-an 385  df-vd1 39103
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator