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Theorem biadanid 30239
Description: Deduction associated with biadani 819. Add a conjunction to an equivalence. (Contributed by Thierry Arnoux, 16-Jun-2024.)
Hypotheses
Ref Expression
biadanid.1 ((𝜑𝜓) → 𝜒)
biadanid.2 ((𝜑𝜒) → (𝜓𝜃))
Assertion
Ref Expression
biadanid (𝜑 → (𝜓 ↔ (𝜒𝜃)))

Proof of Theorem biadanid
StepHypRef Expression
1 biadanid.1 . . 3 ((𝜑𝜓) → 𝜒)
2 biadanid.2 . . . . . 6 ((𝜑𝜒) → (𝜓𝜃))
32biimpa 480 . . . . 5 (((𝜑𝜒) ∧ 𝜓) → 𝜃)
43an32s 651 . . . 4 (((𝜑𝜓) ∧ 𝜒) → 𝜃)
51, 4mpdan 686 . . 3 ((𝜑𝜓) → 𝜃)
61, 5jca 515 . 2 ((𝜑𝜓) → (𝜒𝜃))
72biimpar 481 . . 3 (((𝜑𝜒) ∧ 𝜃) → 𝜓)
87anasss 470 . 2 ((𝜑 ∧ (𝜒𝜃)) → 𝜓)
96, 8impbida 800 1 (𝜑 → (𝜓 ↔ (𝜒𝜃)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399
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 210  df-an 400
This theorem is referenced by:  ist0cld  31186
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