Theorem biantrud 298

Description: A wff is equivalent to its conjunction with truth. (Contributed by NM, 2-Aug-1994.) (Proof shortened by Wolf Lammen, 23-Oct-2013.)

Hypothesis

Ref	Expression
biantrud.1	⊢ (𝜑 → 𝜓)

Assertion

Ref	Expression
biantrud	⊢ (𝜑 → (𝜒 ↔ (𝜒 ∧ 𝜓)))

Proof of Theorem biantrud

Step	Hyp	Ref	Expression
1		biantrud.1	. 2 ⊢ (𝜑 → 𝜓)
2		iba 294	. 2 ⊢ (𝜓 → (𝜒 ↔ (𝜒 ∧ 𝜓)))
3	1, 2	syl 14	1 ⊢ (𝜑 → (𝜒 ↔ (𝜒 ∧ 𝜓)))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103

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

This theorem is referenced by:  posng  4430  elrnmpt1  4603  fliftf  5459  elxp7  5817  eroveu  6220  reapltxor  7689  divap0b  7771  nnle1eq1  8063  nn0le0eq0  8316  nn0lt10b  8428  ioopos  8973  nndivdvds  10201  dvdsmultr2  10235