Theorem 3anbi1d 1256

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

Hypothesis

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

Assertion

Ref	Expression
3anbi1d	⊢ (φ → ((ψ ∧ θ ∧ τ) ↔ (χ ∧ θ ∧ τ)))

Proof of Theorem 3anbi1d

Step	Hyp	Ref	Expression
1		3anbi1d.1	. 2 ⊢ (φ → (ψ ↔ χ))
2		biidd 228	. 2 ⊢ (φ → (θ ↔ θ))
3	1, 2	3anbi12d 1253	1 ⊢ (φ → ((ψ ∧ θ ∧ τ) ↔ (χ ∧ θ ∧ τ)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ w3a 934

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

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

This theorem is used by:  vtocl3gaf  2924  opkelins2kg  4252  opkelins3kg  4253  opkelsikg  4265  sikss1c1c  4268  brsi  4762  funsi  5521  brsnsi  5774  fnpprod  5844  ovce  6173