Theorem pm4.71i 635

Description: Inference converting an implication to a biconditional with conjunction. Inference from Theorem *4.71 of [WhiteheadRussell] p. 120.

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		pm4.71 633	. 2 ⊢ ((φ → ψ) ↔ (φ ↔ (φ ⋀ ψ)))
3	1, 2	mpbi 189	1 ⊢ (φ ↔ (φ ⋀ ψ))

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223

This theorem is referenced by:  pm4.45 638  2eu5 1446  imadmrn 3398  map0e 4326  xpsnen 4415  aceq5lem2 4708  infmap2lem1 7521  dfms2 7738  pjima 10015

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 147  df-an 225

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