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Theorem mpbir2and 888
Description: Detach a conjunction of truths in a biconditional. (Contributed by NM, 6-Nov-2011.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
Hypotheses
Ref Expression
mpbir2and.1 (φχ)
mpbir2and.2 (φθ)
mpbir2and.3 (φ → (ψ ↔ (χ θ)))
Assertion
Ref Expression
mpbir2and (φψ)

Proof of Theorem mpbir2and
StepHypRef Expression
1 mpbir2and.1 . . 3 (φχ)
2 mpbir2and.2 . . 3 (φθ)
31, 2jca 518 . 2 (φ → (χ θ))
4 mpbir2and.3 . 2 (φ → (ψ ↔ (χ θ)))
53, 4mpbird 223 1 (φψ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 176   wa 358
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 177  df-an 360
This theorem is referenced by:  fvopab4t  5385
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