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Mathbox for David A. Wheeler |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ifnmfalse | Structured version Visualization version GIF version |
Description: If A is not a member of B, but an "if" condition requires it, then the "false" branch results. This is a simple utility to provide a slight shortening and simplification of proofs versus applying iffalse 4540 directly in this case. (Contributed by David A. Wheeler, 15-May-2015.) |
Ref | Expression |
---|---|
ifnmfalse | ⊢ (𝐴 ∉ 𝐵 → if(𝐴 ∈ 𝐵, 𝐶, 𝐷) = 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nel 3045 | . 2 ⊢ (𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵) | |
2 | iffalse 4540 | . 2 ⊢ (¬ 𝐴 ∈ 𝐵 → if(𝐴 ∈ 𝐵, 𝐶, 𝐷) = 𝐷) | |
3 | 1, 2 | sylbi 217 | 1 ⊢ (𝐴 ∉ 𝐵 → if(𝐴 ∈ 𝐵, 𝐶, 𝐷) = 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1537 ∈ wcel 2106 ∉ wnel 3044 ifcif 4531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nel 3045 df-if 4532 |
This theorem is referenced by: (None) |
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