Theorem biantrurd 494

Description: A wff is equivalent to its conjunction with truth. (Contributed by NM, 1-May-1995.) (Proof shortened by Andrew Salmon, 7-May-2011.)

Hypothesis

Ref	Expression
biantrud.1	⊢ (φ → ψ)

Assertion

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

Proof of Theorem biantrurd

Step	Hyp	Ref	Expression
1		biantrud.1	. 2 ⊢ (φ → ψ)
2		ibar 490	. 2 ⊢ (ψ → (χ ↔ (ψ ∧ χ)))
3	1, 2	syl 15	1 ⊢ (φ → (χ ↔ (ψ ∧ χ)))

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:  3anibar  1123  n0moeu  3563  opkelcokg  4262  opkelimagekg  4272  reiota2  4369  opbrop  4842  funcnv3  5158  fnssresb  5196  dff1o5  5296  dffo3  5423  fconst4  5459  eloprabga  5579  nenpw1pwlem2  6086