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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 3504 | . 2 ⊢ (𝜑 → 𝐴 ∈ V) |
| 3 | ifexd.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
| 4 | 3 | elexd 3504 | . 2 ⊢ (𝜑 → 𝐵 ∈ V) |
| 5 | 2, 4 | ifcld 4572 | 1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ V) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 Vcvv 3480 ifcif 4525 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-if 4526 |
| This theorem is referenced by: ifexg 4575 evlslem3 22104 mhpsclcl 22151 psgnfzto1stlem 33120 prjspnfv01 42634 prjspner01 42635 prjspner1 42636 sge0val 46381 hsphoival 46594 hspmbllem2 46642 |
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