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Theorem biadan2 456
Description: Add a conjunction to an equivalence. (Contributed by Jeff Madsen, 20-Jun-2011.)
Hypotheses
Ref Expression
biadan2.1 (𝜑𝜓)
biadan2.2 (𝜓 → (𝜑𝜒))
Assertion
Ref Expression
biadan2 (𝜑 ↔ (𝜓𝜒))

Proof of Theorem biadan2
StepHypRef Expression
1 biadan2.1 . . 3 (𝜑𝜓)
21pm4.71ri 392 . 2 (𝜑 ↔ (𝜓𝜑))
3 biadan2.2 . . 3 (𝜓 → (𝜑𝜒))
43pm5.32i 454 . 2 ((𝜓𝜑) ↔ (𝜓𝜒))
52, 4bitri 184 1 (𝜑 ↔ (𝜓𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108
This theorem depends on definitions:  df-bi 117
This theorem is referenced by:  elab4g  2886  elpwb  3584  ssdifsn  3719  brab2a  4676  brab2ga  4698  elovmpo  6066  eqop2  6173  elnnnn0  9208  elixx3g  9888  elfzo2  10136  1nprm  12097
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