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Theorem 3anbi13d 1220
Description: Deduction conjoining and adding a conjunct to equivalences. (Contributed by NM, 8-Sep-2006.)
Hypotheses
Ref Expression
3anbi12d.1 (𝜑 → (𝜓𝜒))
3anbi12d.2 (𝜑 → (𝜃𝜏))
Assertion
Ref Expression
3anbi13d (𝜑 → ((𝜓𝜂𝜃) ↔ (𝜒𝜂𝜏)))

Proof of Theorem 3anbi13d
StepHypRef Expression
1 3anbi12d.1 . 2 (𝜑 → (𝜓𝜒))
2 biidd 165 . 2 (𝜑 → (𝜂𝜂))
3 3anbi12d.2 . 2 (𝜑 → (𝜃𝜏))
41, 2, 33anbi123d 1218 1 (𝜑 → ((𝜓𝜂𝜃) ↔ (𝜒𝜂𝜏)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  w3a 896
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105
This theorem depends on definitions:  df-bi 114  df-3an 898
This theorem is referenced by:  3anbi3d  1224  ltxrlt  7143
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