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Theorem pm4.45im 317
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 5 . . 3 (𝜑 → (𝜓𝜑))
21ancli 306 . 2 (𝜑 → (𝜑 ∧ (𝜓𝜑)))
3 simpl 102 . 2 ((𝜑 ∧ (𝜓𝜑)) → 𝜑)
42, 3impbii 117 1 (𝜑 ↔ (𝜑 ∧ (𝜓𝜑)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101
This theorem depends on definitions:  df-bi 110
This theorem is referenced by:  difdif  3069
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