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Mirrors > Home > ILE Home > Th. List > jcab | GIF version |
Description: Distributive law for implication over conjunction. Compare Theorem *4.76 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Wolf Lammen, 27-Nov-2013.) |
Ref | Expression |
---|---|
jcab | ⊢ ((φ → (ψ ∧ χ)) ↔ ((φ → ψ) ∧ (φ → χ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 102 | . . . 4 ⊢ ((ψ ∧ χ) → ψ) | |
2 | 1 | imim2i 12 | . . 3 ⊢ ((φ → (ψ ∧ χ)) → (φ → ψ)) |
3 | simpr 103 | . . . 4 ⊢ ((ψ ∧ χ) → χ) | |
4 | 3 | imim2i 12 | . . 3 ⊢ ((φ → (ψ ∧ χ)) → (φ → χ)) |
5 | 2, 4 | jca 290 | . 2 ⊢ ((φ → (ψ ∧ χ)) → ((φ → ψ) ∧ (φ → χ))) |
6 | pm3.43 534 | . 2 ⊢ (((φ → ψ) ∧ (φ → χ)) → (φ → (ψ ∧ χ))) | |
7 | 5, 6 | impbii 117 | 1 ⊢ ((φ → (ψ ∧ χ)) ↔ ((φ → ψ) ∧ (φ → χ))) |
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: pm4.76 536 pm5.44 833 2eu4 1990 ssconb 3070 ssin 3153 raaan 3321 tfri3 5894 |
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