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Theorem 3anbi12d 1292
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
3anbi12d (𝜑 → ((𝜓𝜃𝜂) ↔ (𝜒𝜏𝜂)))

Proof of Theorem 3anbi12d
StepHypRef Expression
1 3anbi12d.1 . 2 (𝜑 → (𝜓𝜒))
2 3anbi12d.2 . 2 (𝜑 → (𝜃𝜏))
3 biidd 171 . 2 (𝜑 → (𝜂𝜂))
41, 2, 33anbi123d 1291 1 (𝜑 → ((𝜓𝜃𝜂) ↔ (𝜒𝜏𝜂)))
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:  3anbi1d  1295  3anbi2d  1296  fseq1m1p1  9905
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