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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 version of the definition uses only primitive symbols (→ , ¬ , ∀). This is the definition used in Section II.24 of [Church] p. 129 (Definition D12 page 132) (see comment of df-ifp 1056). (Contributed by BJ, 22-Jun-2019.) |
Ref | Expression |
---|---|
dfifp2 | ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ifp 1056 | . 2 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒))) | |
2 | cases2 1040 | . 2 ⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)) ↔ ((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒))) | |
3 | 1, 2 | bitri 276 | 1 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∨ wo 842 if-wif 1055 |
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 208 df-an 397 df-or 843 df-ifp 1056 |
This theorem is referenced by: dfifp3 1058 dfifp5 1060 ifpn 1065 ifpimpda 1071 revwlk 31987 ifpbi2 39342 ifpbi3 39343 ifpbi23 39348 ifpbi1 39353 ifpbi12 39364 ifpbi13 39365 ifpimimb 39380 ifpororb 39381 ifpbibib 39386 frege54cor0a 39719 |
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