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Mirrors > Home > ILE Home > Th. List > biorf | GIF version |
Description: A wff is equivalent to its disjunction with falsehood. Theorem *4.74 of [WhiteheadRussell] p. 121. (Contributed by NM, 23-Mar-1995.) (Proof shortened by Wolf Lammen, 18-Nov-2012.) |
Ref | Expression |
---|---|
biorf | ⊢ (¬ 𝜑 → (𝜓 ↔ (𝜑 ∨ 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | olc 632 | . 2 ⊢ (𝜓 → (𝜑 ∨ 𝜓)) | |
2 | orel1 644 | . 2 ⊢ (¬ 𝜑 → ((𝜑 ∨ 𝜓) → 𝜓)) | |
3 | 1, 2 | impbid2 131 | 1 ⊢ (¬ 𝜑 → (𝜓 ↔ (𝜑 ∨ 𝜓))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 98 ∨ wo 629 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in2 545 ax-io 630 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: biortn 664 pm5.61 708 pm5.55dc 819 euor 1926 eueq3dc 2715 difprsnss 3502 opthprc 4391 frecsuclem3 5990 swoord1 6135 indpi 6440 enq0tr 6532 mulap0r 7606 mulge0 7610 leltap 7615 ap0gt0 7629 |
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