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Mirrors > Home > MPE Home > Th. List > dfifp2 | Structured version Visualization version GIF version |
Description: Alternate definition of the conditional operator for propositions. The value of if-(𝜑, 𝜓, 𝜒) is "if 𝜑 then 𝜓, and if not 𝜑 then 𝜒." This is the definition used in Section II.24 of [Church] p. 129 (Definition D12 page 132) (see comment of df-ifp 1007). (Contributed by BJ, 22-Jun-2019.) |
Ref | Expression |
---|---|
dfifp2 | ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ifp 1007 | . 2 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒))) | |
2 | cases2 1005 | . 2 ⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)) ↔ ((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒))) | |
3 | 1, 2 | bitri 263 | 1 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 if-wif 1006 |
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 196 df-or 384 df-an 385 df-ifp 1007 |
This theorem is referenced by: dfifp3 1009 dfifp5 1011 ifpn 1016 ifpimpda 1022 ifpbi2 36830 ifpbi3 36831 ifpbi23 36836 ifpbi1 36841 ifpbi12 36852 ifpbi13 36853 ifpbi123 36854 ifpimimb 36868 ifpororb 36869 ifpbibib 36874 frege54cor0a 37177 |
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