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Theorem 3anbi3d 1297
Description: Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.)
Hypothesis
Ref Expression
3anbi1d.1 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
3anbi3d (𝜑 → ((𝜃𝜏𝜓) ↔ (𝜃𝜏𝜒)))

Proof of Theorem 3anbi3d
StepHypRef Expression
1 biidd 171 . 2 (𝜑 → (𝜃𝜃))
2 3anbi1d.1 . 2 (𝜑 → (𝜓𝜒))
31, 23anbi13d 1293 1 (𝜑 → ((𝜃𝜏𝜓) ↔ (𝜃𝜏𝜒)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  w3a 963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107
This theorem depends on definitions:  df-bi 116  df-3an 965
This theorem is referenced by:  ceqsex3v  2731  ceqsex4v  2732  ceqsex8v  2734  vtocl3gaf  2758  mob  2869  ordsoexmid  4484  tfr1onlemaccex  6252  tfrcllemaccex  6265  fseq1m1p1  9905  summodc  11183  fsum3  11187  divalglemnn  11649  divalglemeunn  11652  divalglemex  11653  divalglemeuneg  11654
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