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Theorem mpbir3an 1121
Description: Detach a conjunction of truths in a biconditional. (Contributed by NM, 16-Sep-2011.) (Revised by NM, 9-Jan-2015.)
Hypotheses
Ref Expression
mpbir3an.1 𝜓
mpbir3an.2 𝜒
mpbir3an.3 𝜃
mpbir3an.4 (𝜑 ↔ (𝜓𝜒𝜃))
Assertion
Ref Expression
mpbir3an 𝜑

Proof of Theorem mpbir3an
StepHypRef Expression
1 mpbir3an.1 . . 3 𝜓
2 mpbir3an.2 . . 3 𝜒
3 mpbir3an.3 . . 3 𝜃
41, 2, 33pm3.2i 1117 . 2 (𝜓𝜒𝜃)
5 mpbir3an.4 . 2 (𝜑 ↔ (𝜓𝜒𝜃))
64, 5mpbir 144 1 𝜑
Colors of variables: wff set class
Syntax hints:  wb 103  w3a 920
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106
This theorem depends on definitions:  df-bi 115  df-3an 922
This theorem is referenced by:  limon  4265  limom  4362  issmo  5937  1eluzge0  8743  2eluzge1  8745  0elunit  9084  1elunit  9085  4fvwrd4  9227  fzo0to42pr  9306  resqrexlemga  10047
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