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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 3430 | . 2 ⊢ (𝜑 → 𝐴 ∈ V) |
3 | ifexd.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
4 | 3 | elexd 3430 | . 2 ⊢ (𝜑 → 𝐵 ∈ V) |
5 | 2, 4 | ifcld 4469 | 1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 Vcvv 3409 ifcif 4423 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2729 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-sb 2070 df-clab 2736 df-cleq 2750 df-clel 2830 df-v 3411 df-if 4424 |
This theorem is referenced by: ifexg 4472 evlslem3 20856 mhpsclcl 20903 psgnfzto1stlem 30905 prjspnfv01 39993 prjspner01 39994 prjspner1 39995 sge0val 43406 hsphoival 43619 hspmbllem2 43667 |
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