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Theorem bian1d 28478
Description: Adding a superfluous conjunct in a biconditional. (Contributed by Thierry Arnoux, 26-Feb-2017.)
Hypothesis
Ref Expression
bian1d.1 (𝜑 → (𝜓 ↔ (𝜒𝜃)))
Assertion
Ref Expression
bian1d (𝜑 → ((𝜒𝜓) ↔ (𝜒𝜃)))

Proof of Theorem bian1d
StepHypRef Expression
1 bian1d.1 . . . 4 (𝜑 → (𝜓 ↔ (𝜒𝜃)))
21biimpd 217 . . 3 (𝜑 → (𝜓 → (𝜒𝜃)))
32adantld 481 . 2 (𝜑 → ((𝜒𝜓) → (𝜒𝜃)))
4 simpl 471 . . . 4 ((𝜒𝜃) → 𝜒)
54a1i 11 . . 3 (𝜑 → ((𝜒𝜃) → 𝜒))
61biimprd 236 . . 3 (𝜑 → ((𝜒𝜃) → 𝜓))
75, 6jcad 553 . 2 (𝜑 → ((𝜒𝜃) → (𝜒𝜓)))
83, 7impbid 200 1 (𝜑 → ((𝜒𝜓) ↔ (𝜒𝜃)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382
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 195  df-an 384
This theorem is referenced by:  funcnvmptOLD  28638  funcnvmpt  28639
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