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Theorem 3anbi23d 1251
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
3anbi23d (𝜑 → ((𝜂𝜓𝜃) ↔ (𝜂𝜒𝜏)))

Proof of Theorem 3anbi23d
StepHypRef Expression
1 biidd 170 . 2 (𝜑 → (𝜂𝜂))
2 3anbi12d.1 . 2 (𝜑 → (𝜓𝜒))
3 3anbi12d.2 . 2 (𝜑 → (𝜃𝜏))
41, 2, 33anbi123d 1248 1 (𝜑 → ((𝜂𝜓𝜃) ↔ (𝜂𝜒𝜏)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  w3a 924
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 926
This theorem is referenced by:  ltxrlt  7550  dfgcd2  11277
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