Theorem 3anbi3d 1258

Description: Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.)

Hypothesis

Ref	Expression
3anbi1d.1	⊢ (φ → (ψ ↔ χ))

Assertion

Ref	Expression
3anbi3d	⊢ (φ → ((θ ∧ τ ∧ ψ) ↔ (θ ∧ τ ∧ χ)))

Proof of Theorem 3anbi3d

Step	Hyp	Ref	Expression
1		biidd 228	. 2 ⊢ (φ → (θ ↔ θ))
2		3anbi1d.1	. 2 ⊢ (φ → (ψ ↔ χ))
3	1, 2	3anbi13d 1254	1 ⊢ (φ → ((θ ∧ τ ∧ ψ) ↔ (θ ∧ τ ∧ χ)))

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:  ceqsex3v  2898  ceqsex4v  2899  ceqsex8v  2901  vtocl3gaf  2924  mob  3019  ins2keq  4219  ins3keq  4220  sikeq  4242  ceex  6175  elce  6176