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Theorem mpbir3and 1124
Description: Detach a conjunction of truths in a biconditional. (Contributed by Mario Carneiro, 11-May-2014.)
Hypotheses
Ref Expression
mpbir3and.1 (𝜑𝜒)
mpbir3and.2 (𝜑𝜃)
mpbir3and.3 (𝜑𝜏)
mpbir3and.4 (𝜑 → (𝜓 ↔ (𝜒𝜃𝜏)))
Assertion
Ref Expression
mpbir3and (𝜑𝜓)

Proof of Theorem mpbir3and
StepHypRef Expression
1 mpbir3and.1 . . 3 (𝜑𝜒)
2 mpbir3and.2 . . 3 (𝜑𝜃)
3 mpbir3and.3 . . 3 (𝜑𝜏)
41, 2, 33jca 1121 . 2 (𝜑 → (𝜒𝜃𝜏))
5 mpbir3and.4 . 2 (𝜑 → (𝜓 ↔ (𝜒𝜃𝜏)))
64, 5mpbird 165 1 (𝜑𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  w3a 922
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 924
This theorem is referenced by:  ixxss1  9254  ixxss2  9255  ixxss12  9256  ubioc1  9279  lbico1  9280  lbicc2  9333  ubicc2  9334  modqelico  9669  zmodfz  9681  modqmuladdim  9702  addmodid  9707  phicl2  11065
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