Theorem pm4.71i 483

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		pm4.71 481	. 2 ⊢ ((φ → ψ) ↔ (φ ↔ (φ ∧ ψ)))
3	1, 2	mpbi 164	1 ⊢ (φ ↔ (φ ∧ ψ))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196

This theorem is referenced by:  2eu5 1071  imadmrn 2610  map0e 3266  xpsnen 3339  aceq5lem2 3559  infmap2lem1 4951

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

This theorem depends on definitions:  df-bi 128  df-an 198

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