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Theorem pm4.45im 582
Description: Conjunction with implication. Compare Theorem *4.45 of [WhiteheadRussell] p. 119. (Contributed by NM, 17-May-1998.)
Assertion
Ref Expression
pm4.45im (𝜑 ↔ (𝜑 ∧ (𝜓𝜑)))

Proof of Theorem pm4.45im
StepHypRef Expression
1 ax-1 6 . . 3 (𝜑 → (𝜓𝜑))
21ancli 571 . 2 (𝜑 → (𝜑 ∧ (𝜓𝜑)))
3 simpl 471 . 2 ((𝜑 ∧ (𝜓𝜑)) → 𝜑)
42, 3impbii 197 1 (𝜑 ↔ (𝜑 ∧ (𝜓𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382
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 195  df-an 384
This theorem is referenced by:  difdif  3693
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