Theorem pm4.71d 615

Description: Deduction converting an implication to a biconditional with conjunction. Deduction from Theorem *4.71 of [WhiteheadRussell] p. 120. (Contributed by Mario Carneiro, 25-Dec-2016.)

Hypothesis

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

Assertion

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

Proof of Theorem pm4.71d

Step	Hyp	Ref	Expression
1		pm4.71rd.1	. 2 ⊢ (φ → (ψ → χ))
2		pm4.71 611	. 2 ⊢ ((ψ → χ) ↔ (ψ ↔ (ψ ∧ χ)))
3	1, 2	sylib 188	1 ⊢ (φ → (ψ ↔ (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:  ax12bOLD  1690  difin2  3516