Theorem 3anbi12d 1253

Description: Deduction conjoining and adding a conjunct to equivalences. (Contributed by NM, 8-Sep-2006.)

Hypotheses

Ref	Expression
3anbi12d.1	⊢ (φ → (ψ ↔ χ))
3anbi12d.2	⊢ (φ → (θ ↔ τ))

Assertion

Ref	Expression
3anbi12d	⊢ (φ → ((ψ ∧ θ ∧ η) ↔ (χ ∧ τ ∧ η)))

Proof of Theorem 3anbi12d

Step	Hyp	Ref	Expression
1		3anbi12d.1	. 2 ⊢ (φ → (ψ ↔ χ))
2		3anbi12d.2	. 2 ⊢ (φ → (θ ↔ τ))
3		biidd 228	. 2 ⊢ (φ → (η ↔ η))
4	1, 2, 3	3anbi123d 1252	1 ⊢ (φ → ((ψ ∧ θ ∧ η) ↔ (χ ∧ τ ∧ η)))

Syntax hints:   → wi 4   ↔ wb 176   ∧ w3a 934

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936

This theorem is referenced by:  3anbi1d  1256  3anbi2d  1257  enadj  6060