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Mirrors > Home > ILE Home > Th. List > pm5.42 | GIF version |
Description: Theorem *5.42 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) |
Ref | Expression |
---|---|
pm5.42 | ⊢ ((𝜑 → (𝜓 → 𝜒)) ↔ (𝜑 → (𝜓 → (𝜑 ∧ 𝜒)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ibar 301 | . . 3 ⊢ (𝜑 → (𝜒 ↔ (𝜑 ∧ 𝜒))) | |
2 | 1 | imbi2d 230 | . 2 ⊢ (𝜑 → ((𝜓 → 𝜒) ↔ (𝜓 → (𝜑 ∧ 𝜒)))) |
3 | 2 | pm5.74i 180 | 1 ⊢ ((𝜑 → (𝜓 → 𝜒)) ↔ (𝜑 → (𝜓 → (𝜑 ∧ 𝜒)))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 |
This theorem depends on definitions: df-bi 117 |
This theorem is referenced by: anc2l 327 imdistan 444 |
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