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Theorem bitr3VD 39398
Description: Virtual deduction proof of bitr3 341. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
 1:: ⊢ (   (𝜑 ↔ 𝜓)   ▶   (𝜑 ↔ 𝜓)   ) 2:1,?: e1a 39169 ⊢ (   (𝜑 ↔ 𝜓)   ▶   (𝜓 ↔ 𝜑)   ) 3:: ⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜑 ↔ 𝜒)    ▶   (𝜑 ↔ 𝜒)   ) 4:3,?: e2 39173 ⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜑 ↔ 𝜒)    ▶   (𝜒 ↔ 𝜑)   ) 5:2,4,?: e12 39268 ⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜑 ↔ 𝜒)    ▶   (𝜓 ↔ 𝜒)   ) 6:5: ⊢ (   (𝜑 ↔ 𝜓)   ▶   ((𝜑 ↔ 𝜒) → (𝜓 ↔ 𝜒))   ) qed:6: ⊢ ((𝜑 ↔ 𝜓) → ((𝜑 ↔ 𝜒) → (𝜓 ↔ 𝜒)))
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
bitr3VD ((𝜑𝜓) → ((𝜑𝜒) → (𝜓𝜒)))

Proof of Theorem bitr3VD
StepHypRef Expression
1 id 22 . . 3 ((𝜑𝜓) → (𝜑𝜓))
21bicomd 213 . 2 ((𝜑𝜓) → (𝜓𝜑))
3 id 22 . . 3 ((𝜑𝜒) → (𝜑𝜒))
43bicomd 213 . 2 ((𝜑𝜒) → (𝜒𝜑))
5 biantr 992 . . 3 (((𝜓𝜑) ∧ (𝜒𝜑)) → (𝜓𝜒))
65ex 449 . 2 ((𝜓𝜑) → ((𝜒𝜑) → (𝜓𝜒)))
72, 4, 6syl2im 40 1 ((𝜑𝜓) → ((𝜑𝜒) → (𝜓𝜒)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196 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 This theorem is referenced by: (None)
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