Theorem biantrud 493

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 489	. 2 ⊢ (ψ → (χ ↔ (χ ∧ ψ)))
3	1, 2	syl 15	1 ⊢ (φ → (χ ↔ (χ ∧ ψ)))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358

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

This theorem is referenced by:  ssofss  4077  eqtfinrelk  4487  ovmpt2x  5713  clos1induct  5881