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Mirrors > Home > MPE Home > Th. List > ifexg | Structured version Visualization version GIF version |
Description: Existence of the conditional operator (closed form). (Contributed by NM, 21-Mar-2011.) (Proof shortened by BJ, 1-Sep-2022.) |
Ref | Expression |
---|---|
ifexg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → if(𝜑, 𝐴, 𝐵) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 484 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐴 ∈ 𝑉) | |
2 | simpr 486 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐵 ∈ 𝑊) | |
3 | 1, 2 | ifexd 4535 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → if(𝜑, 𝐴, 𝐵) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∈ wcel 2107 Vcvv 3444 ifcif 4487 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3446 df-if 4488 |
This theorem is referenced by: fsuppmptif 9340 cantnfp1lem1 9619 cantnfp1lem3 9621 symgextfv 19205 pmtrfv 19239 marrepeval 21928 gsummatr01lem3 22022 stdbdmetval 23886 stdbdxmet 23887 ellimc2 25257 cdleme31fv 38899 |
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