pm4.71rd - New Foundations Explorer

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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)))

Colors of variables: wff setvar class

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

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8

This theorem depends on definitions: df-bi 177 df-an 360

This theorem is referenced by: 2reu5 3044 ralss 3332 rexss 3333 nnsucelrlem2 4425 dfco2a 5081 feu 5242 funbrfv2b 5362 dffn5 5363 fnrnfv 5364 fniniseg 5371 eqfnfv2 5393 dff4 5421 dff13 5471 nmembers1lem3 6270

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