Theorem pm4.71i 613

Description: Inference converting an implication to a biconditional with conjunction. Inference from Theorem *4.71 of [WhiteheadRussell] p. 120. (Contributed by NM, 4-Jan-2004.)

Hypothesis

Ref	Expression
pm4.71i.1	⊢ (φ → ψ)

Assertion

Ref	Expression
pm4.71i	⊢ (φ ↔ (φ ∧ ψ))

Proof of Theorem pm4.71i

Step	Hyp	Ref	Expression
1		pm4.71i.1	. 2 ⊢ (φ → ψ)
2		pm4.71 611	. 2 ⊢ ((φ → ψ) ↔ (φ ↔ (φ ∧ ψ)))
3	1, 2	mpbi 199	1 ⊢ (φ ↔ (φ ∧ 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:  pm4.24  624  pm4.45  669  anabs1  783  2eu5  2288  brres  4950  dff1o2  5292  map0e  6024  nchoicelem18  6307