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Mirrors > Home > NFE Home > Th. List > pm4.45im | GIF version |
Description: Conjunction with implication. Compare Theorem *4.45 of [WhiteheadRussell] p. 119. (Contributed by NM, 17-May-1998.) |
Ref | Expression |
---|---|
pm4.45im | ⊢ (φ ↔ (φ ∧ (ψ → φ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1 6 | . . 3 ⊢ (φ → (ψ → φ)) | |
2 | 1 | ancli 534 | . 2 ⊢ (φ → (φ ∧ (ψ → φ))) |
3 | simpl 443 | . 2 ⊢ ((φ ∧ (ψ → φ)) → φ) | |
4 | 2, 3 | impbii 180 | 1 ⊢ (φ ↔ (φ ∧ (ψ → φ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 |
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 177 df-an 360 |
This theorem is referenced by: difdif 3393 |
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