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Theorem ifexd 4507
Description: Existence of the conditional operator (deduction form). (Contributed by SN, 26-Jul-2024.)
Hypotheses
Ref Expression
ifexd.1 (𝜑𝐴𝑉)
ifexd.2 (𝜑𝐵𝑊)
Assertion
Ref Expression
ifexd (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ V)

Proof of Theorem ifexd
StepHypRef Expression
1 ifexd.1 . . 3 (𝜑𝐴𝑉)
21elexd 3449 . 2 (𝜑𝐴 ∈ V)
3 ifexd.2 . . 3 (𝜑𝐵𝑊)
43elexd 3449 . 2 (𝜑𝐵 ∈ V)
52, 4ifcld 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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