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Mirrors > Home > MPE Home > Th. List > ifexd | Structured version Visualization version GIF version |
Description: Existence of the conditional operator (deduction form). (Contributed by SN, 26-Jul-2024.) |
Ref | Expression |
---|---|
ifexd.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
ifexd.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
Ref | Expression |
---|---|
ifexd | ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ifexd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | 1 | elexd 3467 | . 2 ⊢ (𝜑 → 𝐴 ∈ V) |
3 | ifexd.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
4 | 3 | elexd 3467 | . 2 ⊢ (𝜑 → 𝐵 ∈ V) |
5 | 2, 4 | ifcld 4536 | 1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 Vcvv 3447 ifcif 4490 |
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 3449 df-if 4491 |
This theorem is referenced by: ifexg 4539 evlslem3 21513 mhpsclcl 21560 psgnfzto1stlem 32005 prjspnfv01 41009 prjspner01 41010 prjspner1 41011 sge0val 44697 hsphoival 44910 hspmbllem2 44958 |
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