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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 3449 | . 2 ⊢ (𝜑 → 𝐴 ∈ V) |
3 | ifexd.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
4 | 3 | elexd 3449 | . 2 ⊢ (𝜑 → 𝐵 ∈ V) |
5 | 2, 4 | ifcld 4505 | 1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 Vcvv 3429 ifcif 4459 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1542 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-v 3431 df-if 4460 |
This theorem is referenced by: ifexg 4508 evlslem3 21300 mhpsclcl 21347 psgnfzto1stlem 31375 prjspnfv01 40469 prjspner01 40470 prjspner1 40471 sge0val 43885 hsphoival 44098 hspmbllem2 44146 |
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