Theorem pm4.71rd 616

Description: Deduction converting an implication to a biconditional with conjunction. Deduction from Theorem *4.71 of [WhiteheadRussell] p. 120. (Contributed by NM, 10-Feb-2005.)

Hypothesis

Ref	Expression
pm4.71rd.1	⊢ (φ → (ψ → χ))

Assertion

Ref	Expression
pm4.71rd	⊢ (φ → (ψ ↔ (χ ∧ ψ)))

Proof of Theorem pm4.71rd

Step	Hyp	Ref	Expression
1		pm4.71rd.1	. 2 ⊢ (φ → (ψ → χ))
2		pm4.71r 612	. 2 ⊢ ((ψ → χ) ↔ (ψ ↔ (χ ∧ ψ)))
3	1, 2	sylib 188	1 ⊢ (φ → (ψ ↔ (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:  2reu5  3045  ralss  3333  rexss  3334  nnsucelrlem2  4426  dfco2a  5082  feu  5243  funbrfv2b  5363  dffn5  5364  fnrnfv  5365  fniniseg  5372  eqfnfv2  5394  dff4  5422  dff13  5472  nmembers1lem3  6271